Quardatic equation
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This document defines a quadratic equation as one that contains an unknown quantity squared but not to higher powers, and provides the standard form ax^2 + bx + c = 0. It lists three methods for solving quadratic equations: factorization, completing the square, and the quadratic formula. An example problem is worked out step-by-step to show solving a quadratic equation using factorization.
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This document discusses solving quadratic equations by different methods such as factorisation, using the quadratic formula, and completing the square. It explains how to determine the number of solutions a quadratic equation has based on the sign of its discriminant, namely:
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2) Using the square root property to solve equations of the form u^2 = d by taking the square root of both sides.
3) Completing the square to transform quadratic equations into perfect square trinomials that can be factorized.
4) Applying the quadratic formula to solve any quadratic equation in the form ax^2 + bx + c = 0.
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This document outlines key concepts about quadratic expressions and equations. It discusses (1) the form of quadratic expressions as ax2 + bx + c and their parabolic graphs; (2) methods for solving quadratic equations, including factorization, completing the square, and the quadratic formula; and (3) determining the nature of roots and solving equations that reduce to quadratics. Examples are provided to illustrate these concepts and methods. The learning outcomes focus on completing the square to graph quadratics and using appropriate techniques to solve quadratic equations.
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This document provides key points about quadratic equations:
1) The general form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
2) A number α is a root if it satisfies the equation when substituted for x. The roots are the same as the zeroes.
3) The discriminant, D = b2 - 4ac, determines the nature of the roots - positive D means two real roots, zero D means equal real roots, and negative D means no real roots.
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Quadratic equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
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2. x² - 8x + 3 = 0
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- The general form of a quadratic function is f(x) = ax2 + bx + c.
- Key aspects that must be identified before sketching the graph are the nature of the turning point, y-intercept, x-intercepts, and axis of symmetry.
- Quadratic equations can be solved by factorizing, completing the square, or using the quadratic formula. The discriminant determines the nature of the roots.
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How to factor powerpoint
To factor an expression of the form ax^2 + bx + c:
1) Write the expression in the form ax^2 + bx + c and identify a, b, and c.
2) Multiply a and c to find ac and find the greatest common factor of a and c.
3) Find a pair of numbers that multiply to ac and add to b.
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Quadratic equation of class 10
This document provides key points about quadratic equations:
1) The general form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
2) A number α is a root if it satisfies the equation when substituted for x. The roots are the same as the zeroes.
3) The discriminant, D = b2 - 4ac, determines the nature of the roots - positive D means two real roots, zero D means equal real roots, and negative D means no real roots.
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The document discusses quadratic equations and how to solve them by completing the square. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a ≠ 0. It provides examples of perfect square trinomials and how to create them by squaring half the coefficient of the linear term. The method of completing the square to factorize quadratics is explained in multiple steps: taking half the coefficient of the linear term, squaring it, adding and subtracting it to group the first three terms into a perfect square trinomial, and then factorizing. Examples are provided to illustrate solving quadratic equations using this method.
Quadratic equations
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
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The document provides information about trigonometric identities of the form sin(A+B) and double angle formulae. It includes:
1) Trigonometric identity formulas for sin(A+B), sin(A-B), cos(A+B), and cos(A-B).
2) Examples of using the identity formulas to simplify trigonometric expressions and prove identities.
3) Double angle formulas for sin(2A), cos(2A) and their uses in finding exact trig values and solving trig equations.
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- The general form of a quadratic function is f(x) = ax2 + bx + c.
- Key aspects that must be identified before sketching the graph are the nature of the turning point, y-intercept, x-intercepts, and axis of symmetry.
- Quadratic equations can be solved by factorizing, completing the square, or using the quadratic formula. The discriminant determines the nature of the roots.
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- It provides examples of using trigonometric formulae to find unknown sides and angles, including solving trigonometric equations involving double angles.
- Three-dimensional trigonometry is also introduced, defining the angle between two planes and an example problem of finding unknown angles and lengths in a pyramid.
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The document discusses combinations and the binomial theorem. It provides formulas for combinations and explains how to use Pascal's triangle to determine the coefficients in binomial expansions. Examples are worked out expanding (2x + 1)^4, (x - 2y)^3, and finding the coefficient of x^4 in (2x - 7)^9. The key points are that combinations involve choosing objects without regard to order, while permutations consider order; Pascal's triangle determines the coefficients in binomial expansions; and the powers of the terms follow a pattern of the first term decreasing and the second increasing.
How to factor powerpoint
To factor an expression of the form ax^2 + bx + c:
1) Write the expression in the form ax^2 + bx + c and identify a, b, and c.
2) Multiply a and c to find ac and find the greatest common factor of a and c.
3) Find a pair of numbers that multiply to ac and add to b.
4) Write the factored expression by grouping the terms with the numbers found in step 3.
1627 simultaneous equations and intersections
- The document is a demo disc for teaching A-Level maths modules C1 and C2, containing sample slides from 51 presentations covering topics like roots, surds, differentiation, integration, and trigonometry.
- The slides demonstrate key concepts, worked examples, and exercises for students, with explanations of important results and places where students should take notes or do exercises independently.
- The disc is intended to help both students learn and teachers deliver content for the AS core maths modules.
Graphs Of Equations
This document discusses key concepts related to graphing linear and quadratic equations:
- It explains how to graph linear equations in the form of y=mx+c by plotting points based on the slope (m) and y-intercept (c).
- It shows how to find the equation of a line given two points on the line by calculating the slope.
- It describes how the shape of a quadratic graph (y=ax2+bx+c) depends on the values of a, b, and c, and how to find the vertex of a parabola.
- It demonstrates several methods to determine the equation of a quadratic function given points or the vertex.
Solving quadratic equations
The document discusses solving quadratic equations by factoring. It begins with an example problem and explanation of the zero factor property. It then outlines the steps to solve quadratic equations by factoring: 1) write the equation in standard form, 2) factor completely, 3) set each factor equal to 0, 4) solve each equation, and 5) check solutions. Several example problems are worked through demonstrating this process. The document concludes with examples of using factoring to solve real-world problems involving quadratic equations.
Quadratic Equation
A quick intro. to Quadratic Equation.
Few examples have also been discussed to make the student grasp the concept easily.
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This document provides instructions for students to complete algebra warm up problems involving graphing lines from equations. It outlines three methods for graphing lines: using the slope and y-intercept, using a T-chart, and manipulating equations into slope-intercept form. Students are informed they will be tested on using both the slope-intercept method and the T-chart method for graphing lines from equations. They are also told they will need to learn how to manipulate equations in standard form and point-slope form into slope-intercept form.
Jt mathpowerpoint
The document provides examples and steps for factoring different types of polynomials:
1) It shows how to factor binomials that are the difference of squares, like x^2 - 9, by finding the roots.
2) It demonstrates factoring a perfect square trinomial by identifying the pattern a^2 + 2ab + b^2 and grouping terms.
3) It presents the formula for factoring the sum or difference of two cubes, like 8x^3 + 27, by treating it as (a)^3 + (b)^3 and grouping the terms.
16.2 Solving by Factoring
This document provides an overview of solving quadratic equations by factoring. It begins with the standard form of a quadratic equation and explains the zero factor property. Examples are provided to demonstrate factoring quadratic equations and setting each factor equal to zero to solve. The steps for solving a quadratic equation by factoring are outlined. Additional examples demonstrate solving real world application problems involving quadratic equations.
6.3 solving by factoring
This document provides instruction on solving quadratic equations by factoring. It begins by explaining what factoring is and how to factor trinomials of the form x^2 + bx + c. Examples are given of factoring different types of trinomials, including those where b is negative or c is negative. The document then explains how to use the zero product property to solve quadratic equations after factoring them into binomial form. Several examples of solving equations by factoring are shown. Finally, applications of factoring quadratic equations to area word problems are presented.
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Main Java[All of the Base Concepts}.docx
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to deliver Powerpoint Presentations.pptx
"How to make and deliver dynamic presentations by making it more interactive to captivate your audience attention"
How to Setup Warehouse & Location in Odoo 17 Inventory
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South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)Academy of Science of South Africa
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.Chapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptx
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Your Skill Boost Masterclass: Strategies for Effective Upskilling
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Walmart Business+ and Spark Good for Nonprofits.pdf
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
How to Setup Warehouse & Location in Odoo 17 Inventory
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South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)
Chapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering.pptx
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptx
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Quardatic equation
- 2. definition: An equation which contains the square of unknown (variable) quantity but not high power are called quadratic equation or an equation of second degree.
- 3. Formula: Its formula is: X= −𝑏± 𝑏2−4𝑎𝑐 2𝑎 Equation: ax²+bx+c=0
- 4. methods 1. By factorization 2. By completing square 3. By quadratic formula
- 5. Example: 3x²-6x=x+20 Standard form is 3x²-6x=x + 20 Here; a=3 b=-7 c=-20 and ac=3×-20=-60 As -12+5=-7 and -12 × 5=-60 Equation can be written as: 3x²-12x+5x-20=0 3x(x-4)+(x-40)=0 Answer: x-4=0 , 3x+5=0 S . S = {4,-3/5}