5.2factoringquads
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1. This document outlines the steps to factor quadratic expressions. It explains how to factor expressions where the leading coefficient is 1 and where the leading coefficient is not 1. The steps include listing the factors of terms, finding factors whose sum is the coefficient, and organizing the factors into a box form to pull out the greatest common factors.
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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 Formula
The document discusses solving quadratic equations using the quadratic formula. It provides the general form of a quadratic equation as ax2 + bx + c = 0 and introduces the quadratic formula as x = (-b ± √(b2 - 4ac))/2a. Several examples of using the quadratic formula to solve quadratic equations are shown step-by-step with explanations of each step. The examples illustrate determining the a, b, and c coefficients and performing the calculations to find the roots of the equations.
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1. The document describes techniques for integrating trigonometric functions using trigonometric substitution and identities involving sine, cosine, tangent, and secant.
2. Trigonometric substitution involves redefining the variable in terms of a trigonometric function, unlike traditional substitution which defines a new variable.
3. The techniques are demonstrated through examples such as finding antiderivatives of √9-x^2/x^2 and √x^2+4/x^2.
16.1 Solving Quadratics by square roots
This document introduces the square root property and method for solving quadratic equations using square roots. It provides 3 steps: 1) isolate the term being squared, 2) take the square root of both sides, and 3) solve the remaining equation if needed. Examples are shown of solving equations of the form x^2=a, (x-b)^2=c, and (ax-b)^2=c by taking the square root and solving the resulting linear equations. It is noted that some quadratic equations using this method may have no real solutions.
Exponential Generating function
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1.4 Quadratic Equations
This document summarizes key concepts about quadratic equations, including:
- Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
- Completing the square involves manipulating the equation into a perfect square trinomial form.
- The quadratic formula provides the solutions to any quadratic equation in standard form.
- Cubic equations that are the sum or difference of cubes can be factored and solved.
- Literal quadratic equations can be solved for a specified variable using techniques like the square root property or quadratic formula.
- The discriminant determines whether the solutions to a quadratic equation are rational, irrational, or complex numbers.
The 2 Goldbach's Conjectures with Proof
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
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 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 Formula
The document discusses solving quadratic equations using the quadratic formula. It provides the general form of a quadratic equation as ax2 + bx + c = 0 and introduces the quadratic formula as x = (-b ± √(b2 - 4ac))/2a. Several examples of using the quadratic formula to solve quadratic equations are shown step-by-step with explanations of each step. The examples illustrate determining the a, b, and c coefficients and performing the calculations to find the roots of the equations.
Trig substitution
1. The document describes techniques for integrating trigonometric functions using trigonometric substitution and identities involving sine, cosine, tangent, and secant.
2. Trigonometric substitution involves redefining the variable in terms of a trigonometric function, unlike traditional substitution which defines a new variable.
3. The techniques are demonstrated through examples such as finding antiderivatives of √9-x^2/x^2 and √x^2+4/x^2.
16.1 Solving Quadratics by square roots
This document introduces the square root property and method for solving quadratic equations using square roots. It provides 3 steps: 1) isolate the term being squared, 2) take the square root of both sides, and 3) solve the remaining equation if needed. Examples are shown of solving equations of the form x^2=a, (x-b)^2=c, and (ax-b)^2=c by taking the square root and solving the resulting linear equations. It is noted that some quadratic equations using this method may have no real solutions.
Exponential Generating function
The document defines exponential generating functions (EGFs) and provides examples of sequences and their corresponding EGFs. It also shows properties of EGFs such as e^x + e^-x = 2(1 + x^2/2! + x^4/4! + ...) and e^x - e^-x = 2(x + x^3/3! + x^5/5! + ...). Finally, it provides problems involving determining sequences from given EGFs and vice versa.
1.4 Quadratic Equations
This document summarizes key concepts about quadratic equations, including:
- Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
- Completing the square involves manipulating the equation into a perfect square trinomial form.
- The quadratic formula provides the solutions to any quadratic equation in standard form.
- Cubic equations that are the sum or difference of cubes can be factored and solved.
- Literal quadratic equations can be solved for a specified variable using techniques like the square root property or quadratic formula.
- The discriminant determines whether the solutions to a quadratic equation are rational, irrational, or complex numbers.
The 2 Goldbach's Conjectures with Proof
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
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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1) Multiplying powers with the same base means adding the exponents
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3) Raising a power to another exponent means multiplying the exponents
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Worked examples applying each law are provided.
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Carmen sanchez
En esta ocasión te presentamos esta información sobre expresiones algebraicas, además de ello ejercicios variados que te pueden ser mucha utilidad
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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.
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The document discusses how to use the quadratic formula to solve quadratic equations. It provides the formula: x = (-b ± √(b2 - 4ac)) / 2a. It then works through examples of writing quadratic equations in standard form (ax2 + bx + c = 0) and using the formula to solve them. Specifically, it solves the equations: 1) 1x2 + 3x - 27 = 0, 2) 2x2 + 7x + 5 = 0, 3) x2 - 2x = 8, and 4) x2 - 7x = 10. It concludes by providing 3 additional equations to solve using the quadratic formula.
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This document discusses the laws of exponents. It defines what exponents mean and provides examples. The key laws of exponents covered are:
1) Multiplying powers with the same base means adding the exponents
2) Dividing powers with the same base means subtracting the exponents
3) Raising a power to another exponent means multiplying the exponents
4) A number with a zero exponent equals one.
Worked examples applying each law are provided.
Solving Quadratic Equations
This document discusses how to solve quadratic equations by factoring, using the quadratic formula, and determining the number and type of roots using the discriminant. Key steps include factoring the equation if possible to set each factor equal to 0 and solve, plugging the coefficients a, b, and c into the quadratic formula if factoring is not possible, and using the discriminant b^2 - 4ac to determine the number and type of roots. Examples are provided to demonstrate each method.
Quadratic formula
The quadratic formula can be derived by completing the square on the standard form quadratic equation ax^2 + bx + c = 0. By completing the square, the equation can be written as (x + b/2a)^2 = (b^2 - 4ac)/4a^2, from which the quadratic formula -b ± √(b^2 - 4ac)/2a can be identified.
Carmen sanchez
En esta ocasión te presentamos esta información sobre expresiones algebraicas, además de ello ejercicios variados que te pueden ser mucha utilidad
Calculus And Its Applications 10th Edition Bittinger Solutions Manual
Full download : https://goo.gl/76kcJS
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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.
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Fatoring Non Perfect Square Trinomial
1. To factor non-perfect square trinomials, determine the factors of the leading term and the factors of the constant term that can be combined to form the middle term.
2. Trinomials of the form x^2 + bx + c can be factored into the product of two binomials by finding factor pairs of the constant term c that have a sum of b.
3. Examples are provided of factoring trinomials of the form a^2 + 10a + 25, m^2 + 10m + 21, and b^2 + 14b + 45.
Algebra 2 Section 0-3
Here are some key patterns to look for when factoring quadratics:
- Factor out the greatest common factor (GCF)
- Use the difference of squares pattern for expressions of the form a2 - b2
- Use the perfect square trinomial pattern for expressions of the form (a + b)2
- For general quadratics, multiply the first and last terms, find factors that sum to the middle term, and group terms
- Be on the lookout for factoring out expressions like (x - a) from each term
Understanding these common patterns is essential for successfully factoring quadratic expressions. Looking for these patterns can help reveal the underlying structure needed to factor the expression.
EPCA_MODULE-2.pptx
To factor a polynomial using greatest common factors (GCF):
1. Find the GCF of the coefficients and of the variables.
2. The GCF is the factored form of the polynomial.
3. Checking the factored form using the distributive property verifies the correct factorization.
Factoring Polynomials
This document provides instructions on factoring polynomials of various forms:
1) It explains how to factor polynomials by finding the greatest common factor (GCF).
2) It describes how to factor trinomials of the form x2 + bx + c by finding two numbers whose sum is b and product is c.
3) It shows how to factor trinomials of the form ax2 + bx + c by finding factors of a, c whose products and sums satisfy the polynomial.
Lecture 03 factoring polynomials good one
This document provides an overview of Chapter 13 from a developmental mathematics textbook. It covers factoring polynomials, including:
- Finding the greatest common factor of polynomials
- Factoring trinomials of the form x^2 + bx + c and ax^2 + bx + c
- Checking factored results by multiplying the factors
The chapter sections are described and examples are provided for each type of factoring technique.
17 integrals of rational functions x
This document discusses integrating rational functions. It begins by defining rational functions as functions of the form P(x)/Q(x) where P and Q are polynomials. It states that integrating rational functions involves decomposing them into simpler fractions using the Rational Decomposition Theorem. Examples are then provided to demonstrate how to decompose rational functions and integrate each term using substitution.
16 partial fraction decompositions x
This document discusses partial fraction decompositions, which are used to integrate rational functions. It explains that a rational function P(x)/Q(x), where P and Q are polynomials, can be broken down into a sum of simpler rational formulas where the denominators are the factors of Q(x) according to the partial fraction decomposition theorem. Two methods are used to find the exact decomposition: evaluating at the roots of the least common denominator, and matching coefficients after expanding. Examples are provided to illustrate decomposing different types of rational functions.
Factoring Quadratic Trinomials
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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.
Chapter 2
This document provides an overview of quadratic equations and inequalities. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a, b, and c are real number constants and a ≠ 0. Examples of quadratic equations are provided. Methods for solving quadratic equations are discussed, including factoring, completing the square, and the quadratic formula. Properties of inequalities are outlined. The chapter also covers solving polynomial and rational inequalities, as well as equations and inequalities involving absolute value. Practice problems are included at the end.
Quadratic eq and discriminant
The document discusses the quadratic formula and how to use it to solve quadratic equations. It introduces the quadratic formula as x = (-b ±√(b2 - 4ac))/2a and explains how to use it to solve equations by plugging in the a, b, and c coefficients. It also defines the discriminant b2 - 4ac and explains what types of roots (real, complex, rational, irrational) an equation will have depending on whether the discriminant is positive, negative, or zero. Several examples are worked through to demonstrate applying the quadratic formula.
factoting quadratic trinomials.pdf
This document provides instruction on factoring quadratic trinomials of the form x^2 + bx + c. It gives examples of listing the factors of c and choosing a pair that sums to b to write the factored form (x + m)(x + n). It also notes that if the factors of c are 1 and c, the trinomial is prime and cannot be factored with integer coefficients. The document uses factoring to find the dimensions of a box given its volume in terms of x.
1a factorization ok
This document provides examples of factorizing polynomials using various factorization techniques. It contains 30 questions with step-by-step solutions demonstrating how to factorize expressions involving single variables, binomials, trinomials, and polynomials using formulas like difference of squares, perfect square trinomials, and grouping. The techniques shown include finding common factors, using factor trees, recognizing patterns, and applying factorization identities.
ch13powerpoint.ppt
The document summarizes Chapter 13 on factoring polynomials. It covers the greatest common factor, factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. It also discusses factoring perfect square trinomials, difference of two squares, and solving quadratic equations by factoring. Examples are provided to demonstrate each technique.
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.
Factoring quadratic trinomial
The document discusses factoring quadratic trinomials. It provides three examples of factoring expressions such as x^2 - 5x - 14, x^2 - 10x + 25, and x^2 + 2x - 35. Each example lists the steps to factor the expression which include finding factors of the constant term that sum to the coefficient of the middle term and then factoring the first term.
Cuaderno+de+integrales
This document provides a methodology for solving definite and indefinite integrals of various types, including simple, logarithmic, exponential, trigonometric, and their inverses. It contains over 40 examples of integrals worked out step-by-step, covering the basic rules for evaluating indefinite integrals of functions like polynomials, trigonometric functions, exponentials, and their inverses.
Factoring and Box Method
This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
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PC Exam 1 study guide
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A continuous function is defined as one that is defined at a point c, where the limit exists and the function approaches the same y-value from both sides of point c. A function may be discontinuous due to an infinite jump, a jump, or a point discontinuity. Rational functions are defined as the quotient of two polynomial functions. They have asymptotes which occur where the denominator is zero, creating either a vertical or horizontal asymptote in the graph. The end behavior of functions can be determined by considering whether they are increasing or decreasing over different intervals.
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Northern Engraving | Modern Metal Trim, Nameplates and Appliance Panels
Northern Engraving | Modern Metal Trim, Nameplates and Appliance Panels
“Temporal Event Neural Networks: A More Efficient Alternative to the Transfor...
“Temporal Event Neural Networks: A More Efficient Alternative to the Transfor...
5.2factoringquads
- 2. Factoring Quadratics Factoring breaks apart something into smaller pieces.
- 3. Factoring Quadratics a=1 1. List all the pairs of factors of the constant term. x2 - 12x - 28
- 4. Factoring Quadratics a=1 1. List all the pairs of factors of the constant term. x2 - 12x - 28 -1, 28 1, -28 -2, 14 2, -14 -4, 7 4, -7
- 5. Factoring Quadratics a=1 1. List all the pairs of factors of the constant term. 2. Pick the pair whose sum is the coefficient of the x term. x2 - 12x - 28 -1, 28 1, -28 -2, 14 2, -14 -4, 7 4, -7
- 6. Factoring Quadratics a=1 1. List all the pairs of factors of the constant term. 2. Pick the pair whose sum is the coefficient of the x term. x2 - 12x - 28 -1, 28 27 1, -28 -27 -2, 14 12 2, -14 -12 -4, 7 3 4, -7 -3
- 7. Factoring Quadratics a=1 1. List all the pairs of factors of the constant term. 2. Pick the pair whose sum is the coefficient of the x term. 3. Write as factors (x + 2)(x - 14).
- 8. Factoring Quadratics a≠1 1. List all the factors of AC. 3x2 - 17x + 10
- 9. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 3x2 - 17x + 10 1, 30 -1, -30 2, 15 -2, -15 3, 10 -3, -10 5, 6 -5, -6
- 10. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3x2 - 17x + 10 1, 30 -1, -30 2, 15 -2, -15 3, 10 -3, -10 5, 6 -5, -6
- 11. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3x2 - 17x + 10 1, 30 31 -1, -30 -31 2, 15 17 -2, -15 -17 3, 10 13 -3, -10 -13 5, 6 11 -5, -6 -11
- 12. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3. Organize into box 3x2 - 17x + 10
- 13. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3. Organize into box 3x2 - 17x + 10 Ax2 1st number times x 2nd number times x C
- 14. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3. Organize into box 3x2 - 17x + 10 3x2 1st number times x 2nd number times x C
- 15. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3. Organize into box 3x2 - 17x + 10 3x2 1st number times x 2nd number times x 10
- 16. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3. Organize into box 3x2 - 17x + 10 3x2 -2x 2nd number times x 10
- 17. Factoring Quadratics a≠1 1. List all the factors of AC. 3(10) = 30 2. Find the pair that adds to get B. 3. Organize into box 3x2 - 17x + 10 3x2 -2x -15x 10
- 18. Factoring Quadratics a≠1 4. Factor each row and column. 3x2 - 17x + 10 3x2 -2x -15x 10
- 19. Factoring Quadratics a≠1 4. Factor each row and column. 3x2 - 17x + 10 3x2 -2x -15x 10 3x
- 20. Factoring Quadratics a≠1 4. Factor each row and column. 3x2 - 17x + 10 3x2 -2x -15x 10 3x 2
- 21. Factoring Quadratics a≠1 4. Factor each row and column. 3x2 - 17x + 10 3x2 -2x -15x 10 3x 2 x
- 22. Factoring Quadratics a≠1 4. Factor each row and column. 3x2 - 17x + 10 3x2 -2x -15x 10 3x -2 x -5
- 23. Factoring Quadratics a≠1 4. Factor each row and column. 3x2 - 17x + 10 3x2 -2x -15x 10 3x -2 x -5 Sign of box closest goes with the factor.
- 24. Factoring Quadratics a≠1 4. Factor each row and column. 5. These new expressions are the factors of the quadratic. 3x2 - 17x + 10 3x2 -2x -15x 10 3x -2 x -5