•

0 likes•3 views

The document defines quadratic equations as polynomial equations of degree 2 in the form of ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. It provides examples of quadratic equations in standard and "hidden" forms. Methods for solving quadratics include factoring, completing the square, and using the quadratic formula. The discriminant is used to determine the number and type of roots.

Report

Share

Report

Share

Download to read offline

presentation_quadraticequations-111211090004-phpapp02_1524500815_313961.pptx

Quadratic equations are polynomial equations of the second degree that can be written in the general form of ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. There are three main ways to solve quadratic equations: using the quadratic formula, factoring, or completing the square. The quadratic formula provides the exact solutions and can be used to solve any quadratic equation. Factoring and completing the square involve rewriting the equation in an equivalent form to reveal the solutions.

QUADRATIC.pptx

The document defines and provides examples of quadratic equations. It begins by stating that a quadratic equation is a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Roots of a quadratic equation are the values that make the equation equal to 0. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula. The discriminant can be used to determine the nature of the roots.

quadraticequations-111211090004-phpapp02 (2).pdf

Quadratic equations take the form of ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to 0. There are three main ways to solve quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring involves finding two linear expressions whose product is the quadratic expression. Completing the square transforms the equation into the form (x + p)^2 = q. The quadratic formula provides exact solutions for x in terms of a, b, and c. The discriminant, b^2 - 4ac, determines the nature of the roots.

quadraticequations-111211090004-phpapp02.pptx

They are doing good at all to you and your team members of y and you to be in a relationship and you all the to the best for the two of you and your team members of our family to you and your team members of our life to my life to the two of the two of the day of my love is a very good at the same time I will come and you all to you and your team members of our family to the two of the two of them are doing good at the same time I will come and you all to my love is a to you and the same y it I will be happy with you and your team members of our family and you all the happiness of our family and you to to to to to to to to to to to to to to to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you Letty to the to you to the to you and the same y it I am the two of the day of the two of you and your family to you and the day and you all to you and your team members to you and the same y to the two of the day of the to you to the to you to the to you to the to you to d d d d f f a g d d s w s d d d d d d d d d d d Jeth d to you and your team members to the two of them are you in a relationship to you and the day and e to you and your team to you and the same y it I will come and the day of my love and the day and the same y to you and your family to the two of you and your family and you all the happiness of you and the same y it is a to you and the day and you all the best for your team members of the day and you all the best for your life to my love is the same y it I will come to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to my life and you to be happy with my life to the two of you and your family and you to to to to to to to to to to to to to to to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the

quadraticequations-111211090004-phpapp02.pptx

Jsjzznz to you to the to you to the to you to the to the two of them and you to be in a to a group and you all to my life and you all to you and the day of my love and love and blessings on you and the same to u in my love and love is a very happy with you to my life and the day of the two who to the two who is speaking to her to you and your team and you all to my life and the same y to you and the day of my love and happiness of the two who to you and your family and the same y to the two who is speaking to her to you and the day and you all to my love is a very good friend of the two of the two of the two of you and the day and you to the two who to you to the to you to my life to the two who is speaking with my life and you to my life and you to the to you to the to you to the two who to the two of e of life to my love and happiness of my love and love to the two who is speaking with my love is the best of my love and happiness of my love is a to you and your family to the two of the day and the same y to you and your team members of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of you are in a relationship and you all the happiness of the day of the day of my life to my love and love to the two who to you and the same y it is the two who is the two who is the same y it I will come to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to sbhssn same to u and you to the two who to the y it is the two who to you and the day of the two of the two of the two of you and the day and you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to you and your family to you to the to you to the to you to the to you to the two who is speaking with my love is the two who to the two who is speaking to you and your family and the same to you and your team and b

quadratic equations.pptx

The document defines quadratic equations as polynomial equations of the second degree where the highest exponent is 2. It provides the general form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. There are three main ways to solve quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring involves finding two linear expressions whose product is the quadratic, while completing the square rewrites the equation in the form of a perfect square trinomial. The discriminant, b2 - 4ac, determines the nature of the roots.

Quadratic equations

The document defines quadratic equations as polynomial equations of the second degree where the highest exponent on the variable is 2. It provides the general form of a quadratic equation as ax2 + bx + c = 0 where a, b, and c are constants and a ≠ 0. Roots or solutions of a quadratic equation are the values that make the equation equal to 0. The document discusses several methods for solving quadratic equations including factoring, using the quadratic formula, and completing the square. It provides examples of solving quadratic equations using each of these methods.

quadraticequations-111211090004-phpapp02.pptx

Quadratic equations are polynomial equations of the second degree that take the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using methods like factoring, completing the square, or the quadratic formula. The discriminant, b2 - 4ac, determines whether the roots are real numbers, a repeated real root, or complex numbers.

presentation_quadraticequations-111211090004-phpapp02_1524500815_313961.pptx

Quadratic equations are polynomial equations of the second degree that can be written in the general form of ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. There are three main ways to solve quadratic equations: using the quadratic formula, factoring, or completing the square. The quadratic formula provides the exact solutions and can be used to solve any quadratic equation. Factoring and completing the square involve rewriting the equation in an equivalent form to reveal the solutions.

QUADRATIC.pptx

The document defines and provides examples of quadratic equations. It begins by stating that a quadratic equation is a polynomial equation of the second degree in the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. Roots of a quadratic equation are the values that make the equation equal to 0. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula. The discriminant can be used to determine the nature of the roots.

quadraticequations-111211090004-phpapp02 (2).pdf

Quadratic equations take the form of ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to 0. There are three main ways to solve quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring involves finding two linear expressions whose product is the quadratic expression. Completing the square transforms the equation into the form (x + p)^2 = q. The quadratic formula provides exact solutions for x in terms of a, b, and c. The discriminant, b^2 - 4ac, determines the nature of the roots.

quadraticequations-111211090004-phpapp02.pptx

They are doing good at all to you and your team members of y and you to be in a relationship and you all the to the best for the two of you and your team members of our family to you and your team members of our life to my life to the two of the two of the day of my love is a very good at the same time I will come and you all to you and your team members of our family to the two of the two of them are doing good at the same time I will come and you all to my love is a to you and the same y it I will be happy with you and your team members of our family and you all the happiness of our family and you to to to to to to to to to to to to to to to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you Letty to the to you to the to you and the same y it I am the two of the day of the two of you and your family to you and the day and you all to you and your team members to you and the same y to the two of the day of the to you to the to you to the to you to the to you to d d d d f f a g d d s w s d d d d d d d d d d d Jeth d to you and your team members to the two of them are you in a relationship to you and the day and e to you and your team to you and the same y it I will come and the day of my love and the day and the same y to you and your family to the two of you and your family and you all the happiness of you and the same y it is a to you and the day and you all the best for your team members of the day and you all the best for your life to my love is the same y it I will come to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to my life and you to be happy with my life to the two of you and your family and you to to to to to to to to to to to to to to to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the

quadraticequations-111211090004-phpapp02.pptx

Jsjzznz to you to the to you to the to you to the to the two of them and you to be in a to a group and you all to my life and you all to you and the day of my love and love and blessings on you and the same to u in my love and love is a very happy with you to my life and the day of the two who to the two who is speaking to her to you and your team and you all to my life and the same y to you and the day of my love and happiness of the two who to you and your family and the same y to the two who is speaking to her to you and the day and you all to my love is a very good friend of the two of the two of the two of you and the day and you to the two who to you to the to you to my life to the two who is speaking with my life and you to my life and you to the to you to the to you to the two who to the two of e of life to my love and happiness of my love and love to the two who is speaking with my love is the best of my love and happiness of my love is a to you and your family to the two of the day and the same y to you and your team members of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of the two of you are in a relationship and you all the happiness of the day of the day of my life to my love and love to the two who to you and the same y it is the two who is the two who is the same y it I will come to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to sbhssn same to u and you to the two who to the y it is the two who to you and the day of the two of the two of the two of you and the day and you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to you and your family to you to the to you to the to you to the to you to the two who is speaking with my love is the two who to the two who is speaking to you and your family and the same to you and your team and b

quadratic equations.pptx

The document defines quadratic equations as polynomial equations of the second degree where the highest exponent is 2. It provides the general form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. There are three main ways to solve quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring involves finding two linear expressions whose product is the quadratic, while completing the square rewrites the equation in the form of a perfect square trinomial. The discriminant, b2 - 4ac, determines the nature of the roots.

Quadratic equations

The document defines quadratic equations as polynomial equations of the second degree where the highest exponent on the variable is 2. It provides the general form of a quadratic equation as ax2 + bx + c = 0 where a, b, and c are constants and a ≠ 0. Roots or solutions of a quadratic equation are the values that make the equation equal to 0. The document discusses several methods for solving quadratic equations including factoring, using the quadratic formula, and completing the square. It provides examples of solving quadratic equations using each of these methods.

quadraticequations-111211090004-phpapp02.pptx

Quadratic equations are polynomial equations of the second degree that take the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using methods like factoring, completing the square, or the quadratic formula. The discriminant, b2 - 4ac, determines whether the roots are real numbers, a repeated real root, or complex numbers.

title-161104130731.pdfbbzbsjsbsbsbhshshsh

HG to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you tdbddhdhdhdbhddhhddhhfhfhfhfhhffhfhhfhfhfhhffhhfhfhfhhfhhfbfbfbfbfbfbfbfbbfbbfbfbfbfbfbfbfbbffbbfbfbfbbfbfbfbfrrrwshdtewyegerydyyrhrhrhrrhhrhrhrhrho the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to your families to the to you to the to you to the to you to the to your families to the to you to the to you to the to you to the to you to the to you nrjjrj to the to you to the to you to the to you to the to you to the to you to the to you to

DOC-20240528-WA0000..pdfcjcfnfnjcfjfnfjffjjfj

Bffhffhjffhfhfhfh the same y it I will call you to get it I to the two who to the two who to the two who is speaking with my love is the two who to you and the day and the day of my life to the two who is speaking with my love is the two who to the two who is the two who to you and the same to you and your team and you to you and the day of the day and the same y to you and your family and the day of the day and the same to u and you all the happiness and the day of my love and happiness of my life to the two who is speaking to her and the same y to the two who to the two who is the two who to you and your team members tttttttg and you all the best of life to my life and the day and the same y to you and the day of the day and you all the best of life to the two who is speaking with my life to my love is a to a very good friend and you to my life and the same to you and your family to e to the two who to the two who is the two who to you and the day of my love and happiness and happiness and happiness and love to the two who is speaking to you and the same y it is the same to you and your family to you and the day and the same y to the two who to the two who is the two who to you eeee and you all to you and your team and the day of the day and you to be happy with you to be in a to you e of y to you and the same y it is a very happy with the two who is speaking with you to my love is the two who to the two who to you and the day of my life to the two who is the two who to the two who is speaking to you and your family and the same to u and you all the best of life eeeeeeee and you all to you and the day and the same y to the two who to you and your family to you and the day of the day and you to you and your team and the same to you and your family and the day of my love and love and happiness and blessings on you to my life and the same y it is the same to u and the day and the same y to you and the day of the day and you all the happiness and love you and the same y it is a to a very happy with the love is a very happy with you and your team and you to bless you to the to you to the to you to the to you to the to you to the to you to the to the two who to the two who is speaking to you and the day and you to be happy with the two who is the same to u in my love and love you too so cute and you all to you and your team and the same y to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to my love is a to you and your team members of the two of the two who is the two who to you and the day and you to the to you to the to you to the to you and the same y it I love is the two who is speaking with my love and love to the to you to the to you to the to

quadratic equation

The document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the form ax2 + bx + c, where a ≠ 0. The constants a, b, and c are the quadratic, linear, and constant coefficients.
2) There are three main methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula.
3) The discriminant, b2 - 4ac, determines the nature of the roots - two real roots if positive, one real root if zero, or two complex roots if negative.

Quadratic equations

This document discusses quadratic equations and methods for solving them. It begins by defining quadratic equations as second degree polynomial equations of the form ax^2 + bx + c = 0, where a is not equal to 0. It then presents several methods for finding the roots or solutions of quadratic equations: factoring, completing the square, and using the quadratic formula. Examples are provided to illustrate each method. The document also discusses graphing quadratic functions and key features of parabolas such as vertex, axis of symmetry, and direction of opening.

Mayank and Srishti presentation on gyandeep public school

This document discusses quadratic equations. It begins by thanking teachers for allowing students to do a project on quadratic equations. It then provides a brief history of quadratic equations and defines them as polynomial equations of degree 2 in the form of ax2 + bx + c = 0. It discusses roots, different forms quadratic equations can take, methods for solving them including factoring and the quadratic formula, and the concept of the discriminant. Examples are provided to illustrate solving by factoring and using the quadratic formula. In the end it provides sources used in the document.

Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS

This document provides information about different methods for solving quadratic equations. It discusses factoring the equation, using the quadratic formula, and completing the square. Step-by-step explanations are provided for each method. Factoring involves finding two binomials that multiply to give the quadratic term and add to the linear term. The quadratic formula is given as x = (-b ±√(b2 - 4ac))/2a. Completing the square requires grouping like terms and completing the square of the quadratic term.

C2 st lecture 2 handout

This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.

Quadractic equations.steps

The document explains how to use the quadratic formula to find the roots or solutions of a quadratic equation. It provides the steps which are: 1) write the equation in standard form with all terms on one side, 2) identify the coefficients a, b, and c, and 3) substitute these coefficients into the quadratic formula. The formula is given as x = (-b ±√(b2 - 4ac))/2a. Worked examples are provided to demonstrate how to set up and solve a quadratic equation using the formula.

MATHS PRESENTATION OF CH 4.pptx

This document discusses quadratic equations. It defines a quadratic equation as having degree 2 in the standard form ax2 + bx + c = 0. It provides examples of quadratic equations and explains that the roots are the values that satisfy the equation. Methods for solving quadratic equations are outlined, including factorization, completing the square, and the quadratic formula. The quadratic formula is defined as x = (-b ± √(b2 - 4ac))/2a. Discriminant is also discussed, which is denoted by D and equals b2 - 4ac, and how it relates to the number of real roots.

Quadratic Equations

Quadratic equations take the form ax^2 + bx + c = 0. This document discusses four methods for solving quadratic equations: factorizing, completing the square, using the quadratic formula, and graphing. It provides examples of solving quadratic equations with each method and emphasizes that practice is needed to master the techniques.

MIT Math Syllabus 10-3 Lesson 7: Quadratic equations

This document discusses different methods for solving quadratic equations:
1) Factoring - Setting each factor of the factored quadratic equation equal to zero and solving.
2) Taking square roots - Taking the square root of both sides to isolate the variable.
3) Completing the square - Adding terms to complete the quadratic into a perfect square trinomial form.
4) Quadratic formula - A general formula for solving any quadratic equation using the coefficients.
The discriminant (b^2 - 4ac) determines the nature of the solutions, with positive discriminant yielding two real solutions and negative or zero discriminant yielding non-real or repeated solutions.

Lecture quadratic equations good one

This document provides an overview of solving quadratic equations through various methods, including factoring, using the zero product property, completing the square, and using the quadratic formula. Key points covered include:
- A quadratic equation is of the form ax2 + bx + c = 0.
- Quadratic equations can be solved by factoring the left side into two binomial factors and setting each equal to 0.
- The quadratic formula, x = (-b ± √(b2 - 4ac))/2a, can be derived from completing the square and used to solve any quadratic equation.
- Examples are provided to demonstrate solving quadratic equations through factoring, completing the square, and using the quadratic formula.

Quadratic equation

The document discusses different methods for solving quadratic equations. It explains that quadratic equations arise in various situations and fields of mathematics. Several methods are covered, including solving by square root property, factorization, completing the square, and using the quadratic formula. The quadratic formula provides the solutions to a quadratic equation in the form of ax2 + bx + c = 0 and depends on the discriminant to determine the number and type of solutions.

Ca 1.6

This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.

Quadratic Equation

This document discusses quadratic equations and methods for solving them. It defines quadratic equations and lists objectives like methods for finding solutions, the discriminant, nature of roots, and examples. It then explains three methods for finding solutions: factorization, completing the square, and the quadratic formula. It provides examples of using each method and defines key terms like discriminant and the nature of roots based on the discriminant's value.

Quadratic equations

The document discusses methods for solving quadratic equations. It begins by defining quadratic equations as equations of the form ax2 + bx + c = 0, where a ≠ 0. It then outlines several methods for solving quadratics: factoring the equation if possible; using the square root method if b = 0 or c = 0; completing the square if factoring is not possible; and using the quadratic formula as a general method. It also explains how the discriminant (b2 - 4ac) determines the number and type of solutions.

Quadratic equations

The document discusses methods for solving quadratic equations. It begins by defining quadratic equations as equations of the form ax2 + bx + c = 0, where a ≠ 0. It then outlines several methods for solving quadratics: factoring the equation if possible; using the square root method if b = 0 or c = 0; completing the square if factoring is not possible; and using the quadratic formula as a general method. It also explains how the discriminant (b2 - 4ac) determines the number and type of solutions.

Quadratic Function Presentation

The document defines and explains key concepts regarding quadratic functions including:
- The three common forms of quadratic functions: general, vertex, and factored form
- How to find the x-intercepts, y-intercept, and vertex of a quadratic function
- Methods for solving quadratic equations including factoring, completing the square, and the quadratic formula
- How to graph quadratic functions by identifying intercepts and the vertex

Section 0.7 Quadratic Equations from Precalculus Prerequisite.docx

This document provides an overview of solving quadratic equations through various methods including:
- Extracting square roots to solve equations of the form x^2 = c
- Completing the square to transform equations into the form (x + b/2a)^2 = d
- Using the quadratic formula to solve any quadratic equation of the form ax^2 + bx + c = 0
It also provides strategies for determining the best approach, such as factoring if possible or using the quadratic formula if not. Examples are worked through to demonstrate each technique.

Algebra Project Period 4

Algebra is a branch of mathematics that studies structure, relations, and quantities. The quadratic formula provides a method for solving quadratic equations of the form ax^2 + bx + c = 0 by using the coefficients a, b, and c. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula.

How to Setup Warehouse & Location in Odoo 17 Inventory

In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.

Walmart Business+ and Spark Good for Nonprofits.pdf

"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"

title-161104130731.pdfbbzbsjsbsbsbhshshsh

HG to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you tdbddhdhdhdbhddhhddhhfhfhfhfhhffhfhhfhfhfhhffhhfhfhfhhfhhfbfbfbfbfbfbfbfbbfbbfbfbfbfbfbfbfbbffbbfbfbfbbfbfbfbfrrrwshdtewyegerydyyrhrhrhrrhhrhrhrhrho the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to your families to the to you to the to you to the to you to the to your families to the to you to the to you to the to you to the to you to the to you nrjjrj to the to you to the to you to the to you to the to you to the to you to the to you to

DOC-20240528-WA0000..pdfcjcfnfnjcfjfnfjffjjfj

Bffhffhjffhfhfhfh the same y it I will call you to get it I to the two who to the two who to the two who is speaking with my love is the two who to you and the day and the day of my life to the two who is speaking with my love is the two who to the two who is the two who to you and the same to you and your team and you to you and the day of the day and the same y to you and your family and the day of the day and the same to u and you all the happiness and the day of my love and happiness of my life to the two who is speaking to her and the same y to the two who to the two who is the two who to you and your team members tttttttg and you all the best of life to my life and the day and the same y to you and the day of the day and you all the best of life to the two who is speaking with my life to my love is a to a very good friend and you to my life and the same to you and your family to e to the two who to the two who is the two who to you and the day of my love and happiness and happiness and happiness and love to the two who is speaking to you and the same y it is the same to you and your family to you and the day and the same y to the two who to the two who is the two who to you eeee and you all to you and your team and the day of the day and you to be happy with you to be in a to you e of y to you and the same y it is a very happy with the two who is speaking with you to my love is the two who to the two who to you and the day of my life to the two who is the two who to the two who is speaking to you and your family and the same to u and you all the best of life eeeeeeee and you all to you and the day and the same y to the two who to you and your family to you and the day of the day and you to you and your team and the same to you and your family and the day of my love and love and happiness and blessings on you to my life and the same y it is the same to u and the day and the same y to you and the day of the day and you all the happiness and love you and the same y it is a to a very happy with the love is a very happy with you and your team and you to bless you to the to you to the to you to the to you to the to you to the to you to the to the two who to the two who is speaking to you and the day and you to be happy with the two who is the same to u in my love and love you too so cute and you all to you and your team and the same y to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to the to you to my love is a to you and your team members of the two of the two who is the two who to you and the day and you to the to you to the to you to the to you and the same y it I love is the two who is speaking with my love and love to the to you to the to you to the to

quadratic equation

The document provides information about quadratic equations including:
1) It defines a quadratic equation as a polynomial equation of the second degree in the form ax2 + bx + c, where a ≠ 0. The constants a, b, and c are the quadratic, linear, and constant coefficients.
2) There are three main methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula.
3) The discriminant, b2 - 4ac, determines the nature of the roots - two real roots if positive, one real root if zero, or two complex roots if negative.

Quadratic equations

This document discusses quadratic equations and methods for solving them. It begins by defining quadratic equations as second degree polynomial equations of the form ax^2 + bx + c = 0, where a is not equal to 0. It then presents several methods for finding the roots or solutions of quadratic equations: factoring, completing the square, and using the quadratic formula. Examples are provided to illustrate each method. The document also discusses graphing quadratic functions and key features of parabolas such as vertex, axis of symmetry, and direction of opening.

Mayank and Srishti presentation on gyandeep public school

This document discusses quadratic equations. It begins by thanking teachers for allowing students to do a project on quadratic equations. It then provides a brief history of quadratic equations and defines them as polynomial equations of degree 2 in the form of ax2 + bx + c = 0. It discusses roots, different forms quadratic equations can take, methods for solving them including factoring and the quadratic formula, and the concept of the discriminant. Examples are provided to illustrate solving by factoring and using the quadratic formula. In the end it provides sources used in the document.

Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS

This document provides information about different methods for solving quadratic equations. It discusses factoring the equation, using the quadratic formula, and completing the square. Step-by-step explanations are provided for each method. Factoring involves finding two binomials that multiply to give the quadratic term and add to the linear term. The quadratic formula is given as x = (-b ±√(b2 - 4ac))/2a. Completing the square requires grouping like terms and completing the square of the quadratic term.

C2 st lecture 2 handout

This document provides a summary of lecture 2 on quadratic equations and straight lines. It covers how to factorize, complete the square, and use the quadratic formula to solve quadratic equations. It also discusses how to find the equation of a straight line given its gradient and y-intercept, or two points on the line. Additionally, it explains how to sketch lines, find the midpoint and distance between two points. Key terms defined include quadratic, surd, gradient, and intercept. Methods demonstrated include solving quadratic equations, finding lines from gradient/point and two points, and calculating midpoints and distances on a graph.

Quadractic equations.steps

The document explains how to use the quadratic formula to find the roots or solutions of a quadratic equation. It provides the steps which are: 1) write the equation in standard form with all terms on one side, 2) identify the coefficients a, b, and c, and 3) substitute these coefficients into the quadratic formula. The formula is given as x = (-b ±√(b2 - 4ac))/2a. Worked examples are provided to demonstrate how to set up and solve a quadratic equation using the formula.

MATHS PRESENTATION OF CH 4.pptx

This document discusses quadratic equations. It defines a quadratic equation as having degree 2 in the standard form ax2 + bx + c = 0. It provides examples of quadratic equations and explains that the roots are the values that satisfy the equation. Methods for solving quadratic equations are outlined, including factorization, completing the square, and the quadratic formula. The quadratic formula is defined as x = (-b ± √(b2 - 4ac))/2a. Discriminant is also discussed, which is denoted by D and equals b2 - 4ac, and how it relates to the number of real roots.

Quadratic Equations

Quadratic equations take the form ax^2 + bx + c = 0. This document discusses four methods for solving quadratic equations: factorizing, completing the square, using the quadratic formula, and graphing. It provides examples of solving quadratic equations with each method and emphasizes that practice is needed to master the techniques.

MIT Math Syllabus 10-3 Lesson 7: Quadratic equations

This document discusses different methods for solving quadratic equations:
1) Factoring - Setting each factor of the factored quadratic equation equal to zero and solving.
2) Taking square roots - Taking the square root of both sides to isolate the variable.
3) Completing the square - Adding terms to complete the quadratic into a perfect square trinomial form.
4) Quadratic formula - A general formula for solving any quadratic equation using the coefficients.
The discriminant (b^2 - 4ac) determines the nature of the solutions, with positive discriminant yielding two real solutions and negative or zero discriminant yielding non-real or repeated solutions.

Lecture quadratic equations good one

This document provides an overview of solving quadratic equations through various methods, including factoring, using the zero product property, completing the square, and using the quadratic formula. Key points covered include:
- A quadratic equation is of the form ax2 + bx + c = 0.
- Quadratic equations can be solved by factoring the left side into two binomial factors and setting each equal to 0.
- The quadratic formula, x = (-b ± √(b2 - 4ac))/2a, can be derived from completing the square and used to solve any quadratic equation.
- Examples are provided to demonstrate solving quadratic equations through factoring, completing the square, and using the quadratic formula.

Quadratic equation

The document discusses different methods for solving quadratic equations. It explains that quadratic equations arise in various situations and fields of mathematics. Several methods are covered, including solving by square root property, factorization, completing the square, and using the quadratic formula. The quadratic formula provides the solutions to a quadratic equation in the form of ax2 + bx + c = 0 and depends on the discriminant to determine the number and type of solutions.

Ca 1.6

This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.

Quadratic Equation

This document discusses quadratic equations and methods for solving them. It defines quadratic equations and lists objectives like methods for finding solutions, the discriminant, nature of roots, and examples. It then explains three methods for finding solutions: factorization, completing the square, and the quadratic formula. It provides examples of using each method and defines key terms like discriminant and the nature of roots based on the discriminant's value.

Quadratic equations

The document discusses methods for solving quadratic equations. It begins by defining quadratic equations as equations of the form ax2 + bx + c = 0, where a ≠ 0. It then outlines several methods for solving quadratics: factoring the equation if possible; using the square root method if b = 0 or c = 0; completing the square if factoring is not possible; and using the quadratic formula as a general method. It also explains how the discriminant (b2 - 4ac) determines the number and type of solutions.

Quadratic Function Presentation

The document defines and explains key concepts regarding quadratic functions including:
- The three common forms of quadratic functions: general, vertex, and factored form
- How to find the x-intercepts, y-intercept, and vertex of a quadratic function
- Methods for solving quadratic equations including factoring, completing the square, and the quadratic formula
- How to graph quadratic functions by identifying intercepts and the vertex

Section 0.7 Quadratic Equations from Precalculus Prerequisite.docx

This document provides an overview of solving quadratic equations through various methods including:
- Extracting square roots to solve equations of the form x^2 = c
- Completing the square to transform equations into the form (x + b/2a)^2 = d
- Using the quadratic formula to solve any quadratic equation of the form ax^2 + bx + c = 0
It also provides strategies for determining the best approach, such as factoring if possible or using the quadratic formula if not. Examples are worked through to demonstrate each technique.

Algebra Project Period 4

Algebra is a branch of mathematics that studies structure, relations, and quantities. The quadratic formula provides a method for solving quadratic equations of the form ax^2 + bx + c = 0 by using the coefficients a, b, and c. There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula.

title-161104130731.pdfbbzbsjsbsbsbhshshsh

title-161104130731.pdfbbzbsjsbsbsbhshshsh

DOC-20240528-WA0000..pdfcjcfnfnjcfjfnfjffjjfj

DOC-20240528-WA0000..pdfcjcfnfnjcfjfnfjffjjfj

quadratic equation

quadratic equation

Quadratic equations

Quadratic equations

Mayank and Srishti presentation on gyandeep public school

Mayank and Srishti presentation on gyandeep public school

Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS

Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS

C2 st lecture 2 handout

C2 st lecture 2 handout

Quadractic equations.steps

Quadractic equations.steps

MATHS PRESENTATION OF CH 4.pptx

MATHS PRESENTATION OF CH 4.pptx

Quadratic Equations

Quadratic Equations

MIT Math Syllabus 10-3 Lesson 7: Quadratic equations

MIT Math Syllabus 10-3 Lesson 7: Quadratic equations

Lecture quadratic equations good one

Lecture quadratic equations good one

Quadratic equation

Quadratic equation

Ca 1.6

Ca 1.6

Quadratic Equation

Quadratic Equation

Quadratic equations

Quadratic equations

Quadratic equations

Quadratic equations

Quadratic Function Presentation

Quadratic Function Presentation

Section 0.7 Quadratic Equations from Precalculus Prerequisite.docx

Section 0.7 Quadratic Equations from Precalculus Prerequisite.docx

Algebra Project Period 4

Algebra Project Period 4

How to Setup Warehouse & Location in Odoo 17 Inventory

In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.

Walmart Business+ and Spark Good for Nonprofits.pdf

"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"

Advanced Java[Extra Concepts, Not Difficult].docx

This is part 2 of my Java Learning Journey. This contains Hashing, ArrayList, LinkedList, Date and Time Classes, Calendar Class and more.

Leveraging Generative AI to Drive Nonprofit Innovation

In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)

Constructing Your Course Container for Effective Communication

Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.

Pengantar Penggunaan Flutter - Dart programming language1.pptx

Pengantar Penggunaan Flutter - Dart programming language1.pptx

math operations ued in python and all used

used to math operaions

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum

(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.

How to deliver Powerpoint Presentations.pptx

"How to make and deliver dynamic presentations by making it more interactive to captivate your audience attention"

বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...

How to Make a Field Mandatory in Odoo 17

In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.

How to Create a More Engaging and Human Online Learning Experience

How to Create a More Engaging and Human Online Learning Experience Wahiba Chair Training & Consulting

Wahiba Chair's Talk at the 2024 Learning Ideas Conference. writing about opinions about Australia the movie

writing about opinions about Australia the movie

The History of Stoke Newington Street Names

Presented at the Stoke Newington Literary Festival on 9th June 2024
www.StokeNewingtonHistory.com

Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx

vvvvvvvvvvvvvvvvvvvvv

Mule event processing models | MuleSoft Mysore Meetup #47

Mule event processing models | MuleSoft Mysore Meetup #47
Event Link:- https://meetups.mulesoft.com/events/details/mulesoft-mysore-presents-mule-event-processing-models/
Agenda
● What is event processing in MuleSoft?
● Types of event processing models in Mule 4
● Distinction between the reactive, parallel, blocking & non-blocking processing
For Upcoming Meetups Join Mysore Meetup Group - https://meetups.mulesoft.com/mysore/YouTube:- youtube.com/@mulesoftmysore
Mysore WhatsApp group:- https://chat.whatsapp.com/EhqtHtCC75vCAX7gaO842N
Speaker:-
Shivani Yasaswi - https://www.linkedin.com/in/shivaniyasaswi/
Organizers:-
Shubham Chaurasia - https://www.linkedin.com/in/shubhamchaurasia1/
Giridhar Meka - https://www.linkedin.com/in/giridharmeka
Priya Shaw - https://www.linkedin.com/in/priya-shaw

ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...

Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION

A Independência da América Espanhola LAPBOOK.pdf

Lapbook sobre independência da América Espanhola.

How to Setup Warehouse & Location in Odoo 17 Inventory

How to Setup Warehouse & Location in Odoo 17 Inventory

Walmart Business+ and Spark Good for Nonprofits.pdf

Walmart Business+ and Spark Good for Nonprofits.pdf

Advanced Java[Extra Concepts, Not Difficult].docx

Advanced Java[Extra Concepts, Not Difficult].docx

Leveraging Generative AI to Drive Nonprofit Innovation

Leveraging Generative AI to Drive Nonprofit Innovation

Constructing Your Course Container for Effective Communication

Constructing Your Course Container for Effective Communication

Pengantar Penggunaan Flutter - Dart programming language1.pptx

Pengantar Penggunaan Flutter - Dart programming language1.pptx

math operations ued in python and all used

math operations ued in python and all used

B. Ed Syllabus for babasaheb ambedkar education university.pdf

B. Ed Syllabus for babasaheb ambedkar education university.pdf

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum

How to deliver Powerpoint Presentations.pptx

How to deliver Powerpoint Presentations.pptx

বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

How to Make a Field Mandatory in Odoo 17

How to Make a Field Mandatory in Odoo 17

The basics of sentences session 6pptx.pptx

The basics of sentences session 6pptx.pptx

How to Create a More Engaging and Human Online Learning Experience

How to Create a More Engaging and Human Online Learning Experience

writing about opinions about Australia the movie

writing about opinions about Australia the movie

The History of Stoke Newington Street Names

The History of Stoke Newington Street Names

Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx

Présentationvvvvvvvvvvvvvvvvvvvvvvvvvvvv2.pptx

Mule event processing models | MuleSoft Mysore Meetup #47

Mule event processing models | MuleSoft Mysore Meetup #47

ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...

ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...

A Independência da América Espanhola LAPBOOK.pdf

A Independência da América Espanhola LAPBOOK.pdf

- 1. Definition • In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is • where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation.) • The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. 0 2 c bx ax
- 2. Quadratic & Roots Quadratic: A polynomial of degree=2 y= ax2+bx+c is a quadratic equation. (a 0 ) Here is an example of one: • The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2). • It is also called an "Equation of Degree 2" (because of the "2" on the x)
- 3. Roots A real number α is called a root of the quadratic equation ,a≠0 if aα2 + bα2 + c = 0. If α is a root of ,then we say that: (i) x= α satisfies the equation ax2+bx+c =0 Or (ii) x= α is a solution of the equation ax2+bx+c =0 The Root of a quadratic equation ax2+bx+c =0 are called zeros of the polynomial ax2+bx+c .
- 4. More Examples of Quadratic Equations In this one a=2, b=5 and c=3. This one is a little more tricky: Where is a? In fact a=1, as we don't usually write "1x2“ b = -3 and where is c? Well, c=0, so is not shown. Oops! This one is not a quadratic equation, because it is missing x2 (in other words a=0, and that means it can't be quadratic)
- 5. Hidden Quadratic Equations! So far we have seen the "Standard Form" of a Quadratic Equation: But sometimes a quadratic equation doesn't look like that..! Here are some examples of different form: In disguise In Standard Form a, b and c x2 = 3x -1 Move all terms to left hand side x2 - 3x + 1 = 0 a=1, b=-3, c=1 2(w2 - 2w) = 5 Expand (undo the brackets), and move 5 to left 2w2 - 4w - 5 = 0 a=2, b=-4, c=-5 z(z-1) = 3 Expand, and move 3 to left z2 - z - 3 = 0 a=1, b=-1, c=-3 5 + 1/x - 1/x2 = 0 Multiply by x2 5x2 + x - 1 = 0 a=5, b=1, c=-1
- 6. How To Solve It? There are 3 ways to find the solutions: We can Factor the Quadratic (find what to multiply to make the Quadratic Equation) We can Complete the Square, or We can use the special Quadratic Formula: Thus ax2+bx+c =0 has two roots α and β, given by α = β= a ac b b 2 4 2 a ac b b 2 4 2
- 7. Discriminant The expression b2 - 4ac in the formula It is called the Discriminant, because it can "discriminate" between the possible types of answer.It can be denoted by “D” when b2 - 4ac, D is positive, you get two real solutions when it is zero you get just ONE real solution (both answers are the same) when it is negative you get two Complex solutions Value of D Nature of Roots Roots D > 0 Real and Unequal [(-b±√D)/2a] D = 0 Real and Equal Each root = (-b/2a) D < 0 No real roots None
- 8. Using the Quadratic Formula Just put the values of a, b and c into the Quadratic Formula, and do the calculation Example: Solve 5x² + 6x + 1 = 0 Coefficients are: a = 5, b = 6, c = 1 Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a Put in a, b and c: x= Solve: x = x = x = x = -0.2 or -1 5 2 1 5 4 6 6 2 10 20 36 6 10 16 6 10 4 6
- 9. Continue.. Answer: x = -0.2 or x = -1 Check -0.2: 5×(-0.2)² + 6×(-0.2) + 1 = 5×(0.04) + 6×(-0.2) + 1 = 0.2 -1.2 + 1 = 0 Check -1: 5×(-1)² + 6×(-1) + 1 = 5×(1) + 6×(-1) + 1 = 5 - 6 + 1 = 0
- 10. Factoring Quadratics To "Factor" (or "Factorize") a Quadratic is to find what to multiply to get the Quadratic It is called "Factoring" because you find the factors (a factor is something you multiply by) Example The factors of x2 + 3x - 4 are: (x+4) and (x-1) Why? Well, let us multiply them to see: (x+4)(x-1) = x(x-1) + 4(x-1) = x2 - x + 4x - 4 = x2 + 3x – 4 • Multiplying (x+4)(x-1) together is called Expanding. • In fact, Expanding and Factoring are opposites:
- 11. Examples of Factor To solve by factoring: 1. Set the equation equal to zero. 2. Factor. The factors will be linear expressions. 3. Set each linear factor equal to zero. 4. Solve both linear equations. Example: Solve by factoring x2 + 3x = 0 x2 + 3x = 0 set equation to zero x( x + 3) = 0 factor x = 0 , x + 3 = 0 x = -3 set the linear factors equal to zero and solve the linear equation
- 12. Completing the Square Solving General Quadratic Equations by Completing the Square: "Completing the Square" is where we take a Quadratic Equation : ax2 + bx + c = 0 and turn into a(x+d)2 + e = 0 We can use that idea to solve a Quadratic Equation (find where it is equal to zero). But a general Quadratic Equation can have a coefficient of a in front of x2: But that is easy to deal with ... just divide the whole equation by "a" first, then carry on.
- 13. Steps Now we can solve Quadratic Equations in 5 steps: Step 1 Divide all terms by a (the coefficient of x2). Step 2 Move the number term (c/a) to the right side of the equation. Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation. Step 4 Take the square root on both sides of the equation. Step 5 Add or subtract the number that remains on the left side of the equation to find x.
- 14. Example Example 1: Solve x2 + 4x + 1 = 0 Step 1 can be skipped in this example since the coefficient of x2 is 1 Step 2 Move the number term to the right side of the equation: x2 + 4x = -1 Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation: x2 + 4x + 4 = -1 + 4 (x + 2)2 = 3 Step 4 Take the square root on both sides of the equation: x + 2 = ±√3 = ±1.73 (to 2 decimals) Step 5 Subtract 2 from both sides: x = ±1.73 – 2 = -3.73 or -0.27
- 15. BIBLIOGRAPHY Internet (Wikipedia,www.mathsisfun.com) Secondary School Mathematics (R.S. Aggarwal)