Lesson 5.2 honors 2
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This document discusses how to solve quadratic equations by graphing. It defines the terms of a quadratic equation as the quadratic term, linear term, and constant term. Examples are given of identifying these terms in quadratic functions. The document also discusses that quadratic equations can have 0, 1, or 2 real solutions, which are found by determining where the graph crosses the x-axis. Methods for solving quadratic equations graphically include identifying the x-intercepts of the parabolic graph as well as using a table of x-values to plot the graph.
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Dividing Polynomials Slide Share
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Quadratic equations
There are three main methods to solve quadratic equations:
1. Factorization - Factor the left side if possible and use the zero product property to set each factor equal to zero.
2. Completing the square - If factorization is not possible, rearrange the equation into standard form and add a constant to both sides to create a perfect square trinomial. Take the square root of both sides.
3. Quadratic formula - Use the formula ax2 + bx + c = 0, where a ≠ 0 to find the solutions. This method can solve any quadratic equation.
6.4 factoring and solving polynomial equations
The document provides examples and instructions for factoring polynomials of various types, including:
- Trinomials like x^2 - 5x - 12
- Sum and difference of cubes like x^3 + 8 and 8x^3 - 1
- Polynomials with a common monomial factor like 6x^2 + 15x
- Quadratics in the form of au^2 + bu + c
It also discusses using the zero product property to solve polynomial equations by factoring and setting each factor equal to zero.
1.4 Quadratic Equations
This document discusses methods for solving quadratic and cubic equations. It begins by introducing quadratic equations in standard form and methods for solving them, including factoring, completing the square, and using the quadratic formula. It then discusses properties related to the square root and applies them to solving quadratic equations. The document concludes by introducing cubic equations that are the sum or difference of cubes, and provides an example of solving one using factoring.
Solution4
The document discusses deriving the derivative of a quotient function using the quotient rule. It shows applying the quotient rule to the function f(x)=(x^2+3x+2)/(x^2+2) to get the derivative f'(x)=-3(x^2-2)(x^4+4x^2+4)/(x^4+4x^2+4). It then takes the limit as x approaches infinity of this function, simplifying it to lim f(x)=1 as x->∞, demonstrating that the limit exists.
Notes solving polynomial equations
The document provides information about solving polynomial equations. There are three main ways to solve polynomial equations: 1) Using factoring and the zero product property, 2) Using a graphing calculator to graph the equation, and 3) Using synthetic division. The maximum number of solutions a polynomial equation can have is equal to the degree of the polynomial. Examples are provided to demonstrate solving polynomial equations by factoring.
Solving quadratics by graphing
This document discusses how to solve quadratic equations by graphing. It explains that a quadratic equation is of the form y = ax^2 + bx + c, with a being the quadratic term, b the linear term, and c the constant. The number of real solutions is at most two: one solution, two solutions, or no solutions. Solutions are found by setting the equation equal to 0 and finding the x-intercepts of the graph. The graph of a quadratic is a parabola with the roots as the x-intercepts and the vertex as the maximum or minimum point. Quadratic equations can be graphed by making a table of x and y values to plot the parabola.
Synthetic Division
The document provides an explanation and examples of using synthetic division to divide polynomials. Synthetic division allows dividing a polynomial by a divisor of the form (x - k). The process involves writing the coefficients of the dividend in descending order and placing k below. Then, successive multiplication and addition steps provide the coefficients of the quotient polynomial and remainder. Two examples are worked through to demonstrate synthetic division for (2x^3 - 7x^2 - 8x + 16) / (x - 4) and (5x^3 + x^2 - 7) / (x + 1).
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I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
Quadratic equations
There are three main methods to solve quadratic equations:
1. Factorization - Factor the left side if possible and use the zero product property to set each factor equal to zero.
2. Completing the square - If factorization is not possible, rearrange the equation into standard form and add a constant to both sides to create a perfect square trinomial. Take the square root of both sides.
3. Quadratic formula - Use the formula ax2 + bx + c = 0, where a ≠ 0 to find the solutions. This method can solve any quadratic equation.
6.4 factoring and solving polynomial equations
The document provides examples and instructions for factoring polynomials of various types, including:
- Trinomials like x^2 - 5x - 12
- Sum and difference of cubes like x^3 + 8 and 8x^3 - 1
- Polynomials with a common monomial factor like 6x^2 + 15x
- Quadratics in the form of au^2 + bu + c
It also discusses using the zero product property to solve polynomial equations by factoring and setting each factor equal to zero.
1.4 Quadratic Equations
This document discusses methods for solving quadratic and cubic equations. It begins by introducing quadratic equations in standard form and methods for solving them, including factoring, completing the square, and using the quadratic formula. It then discusses properties related to the square root and applies them to solving quadratic equations. The document concludes by introducing cubic equations that are the sum or difference of cubes, and provides an example of solving one using factoring.
Solution4
The document discusses deriving the derivative of a quotient function using the quotient rule. It shows applying the quotient rule to the function f(x)=(x^2+3x+2)/(x^2+2) to get the derivative f'(x)=-3(x^2-2)(x^4+4x^2+4)/(x^4+4x^2+4). It then takes the limit as x approaches infinity of this function, simplifying it to lim f(x)=1 as x->∞, demonstrating that the limit exists.
Notes solving polynomial equations
The document provides information about solving polynomial equations. There are three main ways to solve polynomial equations: 1) Using factoring and the zero product property, 2) Using a graphing calculator to graph the equation, and 3) Using synthetic division. The maximum number of solutions a polynomial equation can have is equal to the degree of the polynomial. Examples are provided to demonstrate solving polynomial equations by factoring.
Solving quadratics by graphing
This document discusses how to solve quadratic equations by graphing. It explains that a quadratic equation is of the form y = ax^2 + bx + c, with a being the quadratic term, b the linear term, and c the constant. The number of real solutions is at most two: one solution, two solutions, or no solutions. Solutions are found by setting the equation equal to 0 and finding the x-intercepts of the graph. The graph of a quadratic is a parabola with the roots as the x-intercepts and the vertex as the maximum or minimum point. Quadratic equations can be graphed by making a table of x and y values to plot the parabola.
Synthetic Division
The document provides an explanation and examples of using synthetic division to divide polynomials. Synthetic division allows dividing a polynomial by a divisor of the form (x - k). The process involves writing the coefficients of the dividend in descending order and placing k below. Then, successive multiplication and addition steps provide the coefficients of the quotient polynomial and remainder. Two examples are worked through to demonstrate synthetic division for (2x^3 - 7x^2 - 8x + 16) / (x - 4) and (5x^3 + x^2 - 7) / (x + 1).
Solving Word Problems Involving Quadratic Equations
This document provides instructions for solving word problems involving quadratic equations. It explains how to write let statements and equations, solve for consecutive integers or areas, and check solutions. Sample problems are worked through, such as finding two consecutive integers whose sum is 13, or the dimensions of a rectangular garden with an area of 27 square units.
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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- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
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.
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polynomial
1) The document discusses polynomials and their properties. It defines what a polynomial is - an expression with terms containing variables and coefficients.
2) It introduces the concept of the degree of a polynomial and defines it as the highest power of the variable terms. Constant polynomials have a degree of zero.
3) The Remainder Theorem is discussed, stating that when dividing a polynomial p(x) by a linear polynomial x-a, the remainder is equal to the value of p(a).
Rational Zeros and Decarte's Rule of Signs
The Rational Zero Theorem provides a method to determine all possible rational zeros of a polynomial function. It states that if p/q is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient. Descartes' Rule of Signs can be used to determine the maximum number of positive and negative real zeros by counting the variations in sign of the polynomial function and its substitution of -x. It provides bounds on the number of positive and negative real zeros that are either the number of variations in sign or less by an even integer. The example demonstrates applying these methods to determine all 16 possible rational zeros and the bounds of 0 positive and either 3 or 1 negative real zeros for the given polynomial.
Quadraticequation
A quadratic equation is a second-order polynomial with terms up to x^2. It has two roots, or solutions, which may be real or complex. The quadratic formula can be used to find the exact numeric values of the roots. A root is a value that makes the quadratic equation equal to zero. A double root occurs when the two roots are equal, making the quadratic a perfect square trinomial.
REDUCIBLE QUADRATIC EQUATIONS
The document discusses solving equations that are reducible to quadratic equations. It explains that a quadratic equation is an equation with a maximum power of the variable being squared. There are three methods for solving quadratic equations: factorization, completing the square, and the quadratic formula. Examples are provided of using factorization and completing the square methods to solve equations reducible to quadratic form. The document also covers forming quadratic equations from word problems and solving them.
Ses 3 quadratic equations
The document discusses several topics in mathematics including:
1) Solving quadratic equations through factorization, graphically, and by taking square roots. Quadratic equations can have two real roots, one repeated root, or no real roots.
2) Drawing linear functions based on the slope and y-intercept. Linear functions can have positive or negative slopes.
3) Drawing the graph of a quadratic function where a is greater than 0, which results in a parabola opening upwards.
4) Defining absolute value as the distance of a number from zero, and that the absolute value of a number must be greater than or equal to zero.
5) Listing several recommended books on statistics.
quadratic equations-1
Okay, let's solve this step-by-step:
1) Given: x = 2 and x = 3 are roots of the equation 3x^2 - 2mx + 2n = 0
2) Substitute x = 2 in the equation: 3(2)^2 - 2m(2) + 2n = 0
=> 12 - 4m + 2n = 0
3) Substitute x = 3 in the equation: 3(3)^2 - 2m(3) + 2n = 0
=> 27 - 6m + 2n = 0
4) Solve the two equations simultaneously to find m and n:
12 - 4m + 2n = 0
27 - 6m +
College Algebra 1.4
This document provides instructions on solving quadratic equations using different methods: factoring, extracting square roots, the quadratic formula, and determining the discriminant. It explains that if the product of two terms is zero, one of the terms must be zero. This is used to solve equations by setting each factor equal to zero. Examples are provided to demonstrate solving quadratic equations by factoring, taking the square root of both sides, and using the discriminant to determine the number of solutions.
Solving Quadratic Equations by Factoring
The document discusses solving quadratic equations by factoring. It provides examples of factoring quadratic expressions to find the solutions to the equations. These include using the zero product rule, factoring a common factor, and factoring a perfect square. It also provides two word problems involving consecutive integers and the Pythagorean theorem and shows how to set up and solve the quadratic equations derived from the word problems.
solving quadratic equations by graphing
how to solve qudratic equations by graphing (with graphs and illustrations step by step) maths lesson
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A quick intro. to Quadratic Equation.
Few examples have also been discussed to make the student grasp the concept easily.
Notes solving polynomials using synthetic division
This document discusses using synthetic division to evaluate polynomials at specific values and factor polynomials. It provides examples of using synthetic division to:
1) Evaluate polynomials like f(x) = x^2 - x + 5 at specific values such as f(-2)
2) Factor polynomials when one factor is known, such as factoring x^3 - 3x^2 - 13x + 15 after determining (x + 3) is a factor
3) Find all zeros of a polynomial by setting each factor equal to 0 after factoring
Step-by-step instructions and additional examples are provided to illustrate the process.
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic Equations
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Remainder theorem
The document discusses the Remainder Theorem, which provides a way to factorize polynomials by dividing them by factors and obtaining a remainder. There are two methods for finding the remainder: long division/evaluation and synthetic division. Evaluation involves substituting the factor value into the polynomial, while synthetic division arranges the coefficients and repeatedly multiplies and adds down the line. The document provides examples of using both methods and notes that synthetic division allows determining the full quotient polynomial.
Test 1 Review
The test this week will cover content from Assignments 0.1 through 0.7. Students are responsible for all content, whether or not they were present for that class. The test questions will be similar to sample questions provided. Review solutions are posted online but will be removed the morning of each test.
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.
6.2 solve quadratic equations by graphing
This document discusses how to solve quadratic equations by graphing. It defines the terms of a quadratic equation as the quadratic term, linear term, and constant term. It shows examples of identifying these terms and finding the solutions or x-intercepts. The document states that a quadratic equation can have 0, 1, or 2 real solutions. It also discusses that the graph of a quadratic is a parabola with roots at the x-intercepts and a vertex as its maximum or minimum point. The axis of symmetry is also identified. Methods for graphing quadratics by making tables or using a graphing calculator are presented.
Bonus math project
This document provides an overview of different methods for solving quadratic equations, including factoring, graphing, using the quadratic formula, and more. It begins by defining the general and standard forms of quadratic equations. It then explains how to write equations in standard form and discusses concepts like the vertex, completing the square, determining zeroes/roots, and the discriminant. Finally, it reviews several methods that can be used to find roots of quadratic and higher-order polynomial equations, such as factoring, graphing, the quadratic formula, synthetic division, and the remainder/factor theorems.
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Solving Word Problems Involving Quadratic Equations
This document provides instructions for solving word problems involving quadratic equations. It explains how to write let statements and equations, solve for consecutive integers or areas, and check solutions. Sample problems are worked through, such as finding two consecutive integers whose sum is 13, or the dimensions of a rectangular garden with an area of 27 square units.
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
GR 8 Math Powerpoint about Polynomial Techniques
-This is a powerpoint inspired by one of Canva displayed presentation.
- This is about Math Polynomials and good for highschoolers presentation for school.
- It consists of 39 pages explaining each of the Polynomial Techniques.
- Good for review or inspired powerpoint.
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.
Ch03 4
The document discusses complex roots of the characteristic equation arising from assuming exponential solutions to a differential equation. It shows that complex roots lead to complex-valued solutions, but linear combinations of solutions can give real-valued solutions in the form of sine and cosine functions. Several examples are worked out to find the general solution of differential equations and determine the time for the solution to drop below a given value based on its graph.
polynomial
1) The document discusses polynomials and their properties. It defines what a polynomial is - an expression with terms containing variables and coefficients.
2) It introduces the concept of the degree of a polynomial and defines it as the highest power of the variable terms. Constant polynomials have a degree of zero.
3) The Remainder Theorem is discussed, stating that when dividing a polynomial p(x) by a linear polynomial x-a, the remainder is equal to the value of p(a).
Rational Zeros and Decarte's Rule of Signs
The Rational Zero Theorem provides a method to determine all possible rational zeros of a polynomial function. It states that if p/q is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient. Descartes' Rule of Signs can be used to determine the maximum number of positive and negative real zeros by counting the variations in sign of the polynomial function and its substitution of -x. It provides bounds on the number of positive and negative real zeros that are either the number of variations in sign or less by an even integer. The example demonstrates applying these methods to determine all 16 possible rational zeros and the bounds of 0 positive and either 3 or 1 negative real zeros for the given polynomial.
Quadraticequation
A quadratic equation is a second-order polynomial with terms up to x^2. It has two roots, or solutions, which may be real or complex. The quadratic formula can be used to find the exact numeric values of the roots. A root is a value that makes the quadratic equation equal to zero. A double root occurs when the two roots are equal, making the quadratic a perfect square trinomial.
REDUCIBLE QUADRATIC EQUATIONS
The document discusses solving equations that are reducible to quadratic equations. It explains that a quadratic equation is an equation with a maximum power of the variable being squared. There are three methods for solving quadratic equations: factorization, completing the square, and the quadratic formula. Examples are provided of using factorization and completing the square methods to solve equations reducible to quadratic form. The document also covers forming quadratic equations from word problems and solving them.
Ses 3 quadratic equations
The document discusses several topics in mathematics including:
1) Solving quadratic equations through factorization, graphically, and by taking square roots. Quadratic equations can have two real roots, one repeated root, or no real roots.
2) Drawing linear functions based on the slope and y-intercept. Linear functions can have positive or negative slopes.
3) Drawing the graph of a quadratic function where a is greater than 0, which results in a parabola opening upwards.
4) Defining absolute value as the distance of a number from zero, and that the absolute value of a number must be greater than or equal to zero.
5) Listing several recommended books on statistics.
quadratic equations-1
Okay, let's solve this step-by-step:
1) Given: x = 2 and x = 3 are roots of the equation 3x^2 - 2mx + 2n = 0
2) Substitute x = 2 in the equation: 3(2)^2 - 2m(2) + 2n = 0
=> 12 - 4m + 2n = 0
3) Substitute x = 3 in the equation: 3(3)^2 - 2m(3) + 2n = 0
=> 27 - 6m + 2n = 0
4) Solve the two equations simultaneously to find m and n:
12 - 4m + 2n = 0
27 - 6m +
College Algebra 1.4
This document provides instructions on solving quadratic equations using different methods: factoring, extracting square roots, the quadratic formula, and determining the discriminant. It explains that if the product of two terms is zero, one of the terms must be zero. This is used to solve equations by setting each factor equal to zero. Examples are provided to demonstrate solving quadratic equations by factoring, taking the square root of both sides, and using the discriminant to determine the number of solutions.
Solving Quadratic Equations by Factoring
The document discusses solving quadratic equations by factoring. It provides examples of factoring quadratic expressions to find the solutions to the equations. These include using the zero product rule, factoring a common factor, and factoring a perfect square. It also provides two word problems involving consecutive integers and the Pythagorean theorem and shows how to set up and solve the quadratic equations derived from the word problems.
solving quadratic equations by graphing
how to solve qudratic equations by graphing (with graphs and illustrations step by step) maths lesson
Quadratic Equation
A quick intro. to Quadratic Equation.
Few examples have also been discussed to make the student grasp the concept easily.
Notes solving polynomials using synthetic division
This document discusses using synthetic division to evaluate polynomials at specific values and factor polynomials. It provides examples of using synthetic division to:
1) Evaluate polynomials like f(x) = x^2 - x + 5 at specific values such as f(-2)
2) Factor polynomials when one factor is known, such as factoring x^3 - 3x^2 - 13x + 15 after determining (x + 3) is a factor
3) Find all zeros of a polynomial by setting each factor equal to 0 after factoring
Step-by-step instructions and additional examples are provided to illustrate the process.
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic Equations
This powerpoint presentation discusses or talks about the topic or lesson System of Equations involving Quadratic Equations. It also discusses and explains the rules, steps and examples of System of Equations involving Quadratic Equations
Remainder theorem
The document discusses the Remainder Theorem, which provides a way to factorize polynomials by dividing them by factors and obtaining a remainder. There are two methods for finding the remainder: long division/evaluation and synthetic division. Evaluation involves substituting the factor value into the polynomial, while synthetic division arranges the coefficients and repeatedly multiplies and adds down the line. The document provides examples of using both methods and notes that synthetic division allows determining the full quotient polynomial.
Test 1 Review
The test this week will cover content from Assignments 0.1 through 0.7. Students are responsible for all content, whether or not they were present for that class. The test questions will be similar to sample questions provided. Review solutions are posted online but will be removed the morning of each test.
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.
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Similar to Lesson 5.2 honors 2
6.2 solve quadratic equations by graphing
This document discusses how to solve quadratic equations by graphing. It defines the terms of a quadratic equation as the quadratic term, linear term, and constant term. It shows examples of identifying these terms and finding the solutions or x-intercepts. The document states that a quadratic equation can have 0, 1, or 2 real solutions. It also discusses that the graph of a quadratic is a parabola with roots at the x-intercepts and a vertex as its maximum or minimum point. The axis of symmetry is also identified. Methods for graphing quadratics by making tables or using a graphing calculator are presented.
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This document provides an overview of different methods for solving quadratic equations, including factoring, graphing, using the quadratic formula, and more. It begins by defining the general and standard forms of quadratic equations. It then explains how to write equations in standard form and discusses concepts like the vertex, completing the square, determining zeroes/roots, and the discriminant. Finally, it reviews several methods that can be used to find roots of quadratic and higher-order polynomial equations, such as factoring, graphing, the quadratic formula, synthetic division, and the remainder/factor theorems.
Graphing quadratic equations
1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
3) An example problem walks through graphing the quadratic equation y = x^2 - 4x by first finding the roots, vertex, and axis of symmetry, and then constructing a table to plot points and graph the parabola.
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
10.4
Here are the steps to solve each equation by graphing:
1) x2 + 5x + 6 = 0
Write in standard form: x2 + 5x + 6 = 0
Graph the related function y = x2 + 5x + 6. The x-intercepts are the solutions, which are -3 and -2.
2) x2 + 8x + 16 = 0
Write in standard form: x2 + 8x + 16 = 0
Graph the related function y = x2 + 8x + 16. The x-intercepts are the solutions, which are -4 and -4.
3) x2 - 2x + 3 = 0
Write in standard
Algebra lesson 4.2 zeroes of quadratic functions
This document provides information about quadratic functions and solving for their zeroes (x-intercepts). It discusses factoring quadratic expressions, using the zero product property to set each factor equal to zero. It also introduces the quadratic formula as a way to solve quadratic equations that are not factorable. There is an example of using the quadratic formula to find the zeroes of the function f(x)=x^2 - 3x - 1. The document concludes with practice problems for students to solve for the zeroes of various quadratic functions.
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The document discusses methods for finding the real solutions of second degree (quadratic) equations. It explains the square root method for equations where the x-term is missing, involving solving for x^2 and taking the square root. It also explains the factoring method, involving factoring the equation into two binomials and setting each equal to 0. The quadratic formula is presented as a general method for solving any second degree equation, and its derivation using completing the square is mentioned.
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This document provides instructions on how to solve various types of equations, including:
- Linear equations by rearranging to isolate the variable
- Equations using the null factor law to find zeros
- Exponential and logarithmic equations by equating exponents or rearranging into logarithmic form
- Logarithmic functions using properties of logarithms
- Using the "solve" command to solve equations on a calculator
- Circular function equations by rearranging to isolate the variable
It also provides examples and exercises to practice these techniques.
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The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations arising from applications like finding the dimensions of a rectangle given its area. Sample problems are worked through step-by-step to find the solutions or roots of quadratic equations.
Solving quadratic equations
The document discusses various methods for solving quadratic equations: factoring, using square roots, and completing the square. It provides examples of using each method to solve equations as well as word problems involving quadratic equations. Factoring involves writing the quadratic as a product of two binomials and setting each factor equal to 0. Using square roots involves taking the square root of both sides of the equation to isolate the variable. Completing the square rewrites the equation by adding or subtracting terms to put it in the form (x+a)2=b and then taking the square root of both sides.
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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.
Solution 3
The document provides the steps to solve a multi-part calculus problem. It involves finding the derivative of two functions, determining the equations of tangent lines to those functions at given points, finding the intersection points of those tangent lines, and using those intersection points to write the equations of three circles with a radius of 5.
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Topic 8 (Writing Equations Of A Straight Lines)
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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.
Solution 3
The document provides the steps to solve a multi-part calculus problem involving derivatives, tangent lines, and circles. It determines the derivative of two functions f(x) and g(x), finds the equations of the tangent lines at specific x-values, identifies the intersection points of the tangent lines, and uses those intersection points as the centers of three circles with a radius of 5 to write the equations of the circles.
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Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where:
a = 3
h = -2
k = 8
So the equation in vertex form is:
y = 3(x - (-2))^2 + 8
y = 3(x + 2)^2 - 4
5.2 solve quadratic equations by graphing.vertex and intercept form
Okay, let's solve this step-by-step:
1) FOIL the expression: y = 3(x + 2)(x + 2) - 4
= 3(x^2 + 4x + 2x + 4) - 4
= 3x^2 + 12x + 12 - 4
= 3x^2 + 12x + 8
2) This is in the vertex form: y = a(x - h)^2 + k
Where: a = 3
h = 0
k = 8
So the equation in vertex form is:
y = 3(x - 0)^2 + 8
Or simply:
y = 3x^2 + 8
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- 1. Solving Quadratic Equation by Graphing Lesson 5.2
- 2. Quadratic Equation y = ax2 + bx + c 2 ax is the quadratic term. bx is the linear term. c is the constant term. The highest exponent is two; therefore, the degree is two.
- 3. Identifying Terms 2 Example f(x)=5x -7x+1 Quadratic term 5x2 Linear term -7x Constant term 1
- 4. Identifying Terms 2 Example f(x) = 4x - 3 Quadratic term 4x Linear term 0 Constant term -3 2
- 5. Identifying Terms Now you try this problem. 2 f(x) = 5x - 2x + 3 5x2 quadratic term linear term constant term -2x 3
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- 7. Solving Equations When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts. These values are also referred to as solutions, zeros, or roots.
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- 9. Identifying Solutions Now you try this problem. f(x) = 2x - x 2 4 2 5 -2 Solutions are 0 and 2. -4
- 10. Graphing Quadratic Equations The graph of a quadratic equation is a parabola. The roots or zeros are the x-intercepts. The vertex is the maximum or minimum point. All parabolas have an axis of symmetry.
- 11. Graphing Quadratic Equations One method of graphing uses a table with arbitrary x-values. Graph y = x2 - 4x 4 2 x y 0 1 2 3 4 0 -3 -4 -3 0 Roots 0 and 4 , Vertex (2, -4) , Axis of Symmetry x = 2 5 -2 -4
- 12. Graphing Quadratic Equations Try this problem y = x2 - 2x - 8. x 4 y -2 -1 1 3 4 2 5 -2 -4 Roots Vertex Axis of Symmetry
- 13. Graphing Quadratic Equations The graphing calculator is also a helpful tool for graphing quadratic equations.