Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 5.3 Factoring Polynomials.
5.3 Factoring and Solving Quadratics (work).notebook
Transcript of 5.3 Factoring and Solving Quadratics (work).notebook
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
The method depends on the form of the equation.
There are several methods available for solving a quadratic equation:
1. By Square Roots2. By Factoring3. By Completing the Square4. By the Quadratic Formula5. By Graphing
5.3 FACTORING QUADRATICS
FACTORING QUADRATIC TRINOMIALS
2. Make a sum/product chart.
5x2 + 17x + 14Example:
3. Divide each number by the leading coefficient.4. Reduce each fraction if possible.5. Denominator = constant or coefficient of first term Numerator = constant or coefficient of last term
1. The expression must be in ascending or descending order.
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Examples: a. x2 + 6x + 8 b. 3x2 - 11x + 6
Examples: c. x2 + 7x - 18 d. 3x2 +10x - 8
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Factor each trinomial.1) x2 - 16x + 39 2) x2 + 2x - 35
3) x2 + 22x + 121 4) x2 - 2x - 63
5) 14x2 - 11x + 2 6) 12x2 + 16x - 3
7) 2x2 + 13x + 6 8) 9x2 - 9x - 28
Practice
Factor each trinomial.1) x2 - 16x + 39 2) x2 + 2x - 35
3) x2 + 22x + 121 4) x2 - 2x - 63
5) 14x2 - 11x + 2 6) 12x2 + 16x - 3
7) 2x2 + 13x + 6 8) 9x2 - 9x - 28
(x - 3)(x - 13) (x + 7)(x - 5)
(x + 11)(x + 11) (x + 7)(x - 9)
(7x - 2)(2x - 1) (2x + 3)(6x - 1)
(2x + 1)(x + 6) (3x + 4)(3x - 7)
Answers
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Special Factoring PatternsFACTORING DIFFERENCE OF SQUARES
x2 4 = (x 2)(x + 2)
4x2 9 = (2x 3)(2x + 3)
x2 49 = (x 7)(x + 7)
64x2 25 = (8x 5)(8x + 5)
a2 b2 =
1.
What is the pattern?
Special Factoring PatternsPERFECT SQUARE TRINOMIALS
x2 + 14x + 49 = (x + 7)2
x2 8x + 16 = (x 4)2
4x2 20x + 25 = (2x 5)2
9x2 + 12x + 4 = (3x + 2)2
a2 2ab + b2 =
2.
What is the pattern?
a2 + 2ab + b2 =
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Practice
Factor completely.1. 4x2 - 121 2. 9x2 - 24x + 16
3. 225 - x2 4. x2 + 10x + 25
5. 10x2 - 13x - 3
Answers
Factor completely.1. 4x2 - 121 2. 9x2 - 24x + 16
3. 225 - x2 4. x2 + 10x + 25
5. 10x2 - 13x - 3
(2x - 11)(2x + 11) (3x - 4)2
(15 - x)(15 + x) (x + 5)2
(2x - 3)(5x + 1)
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
When factoring,ALWAYS look for the GCF first!
Greatest Common Factor the largest factor that divides ALL of the terms
a. 12x2 - 3 b. 7v2 - 42v
FACTOR COMPLETELYc. 5x2 - 45 d. 15x2 + 6x
e. 3x2 - 9x + 6 f. 36x - 48x2 + 24x3
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Practice
Factor completely.1. 12x2 - 3 2. 45x2 + 10x
3. 8x2 - 24x + 18 4. x2 + 5x + 4
5. 6x2 + 13x - 5
Answers
Factor completely.1. 12x2 - 3 2. 45x2 + 10x
3. 8x2 - 24x + 18 4. x2 + 5x + 4
5. 6x2 + 13x - 5
3(2x - 1)(2x + 1) 5x(9x + 2)
2(2x - 3)2 (x + 1)(x + 4)
(2x + 5)(3x - 1)
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
When factoring four terms, use the grouping method.
FACTORING FOUR TERMS
a. x2 - 12x + 3x - 36 b. ra + rb + sa + sb
FACTOR USING THE GROUPING METHOD. c. y2 - 12y - 4y + 48 d. k2 + 3k - 8k - 24
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Practice
Factor completely.1. 2x2y - x + 6xy - 3
2. 6cd2 - 8cd - 9d + 12
3. 2xz - 6xy + 2yz - 6y2
Answers
Factor completely.1. 2x2y - x + 6xy - 3
2. 6cd2 - 8cd - 9d + 12
3. 2xz - 6xy + 2yz - 6y2
(2xy - 1)(x + 3)
2(x + y)(z - 3y)
(2cd - 3)(3d - 4)
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Solving ax2 + bx + c = 0
byFACTORING
The solutions of a quadratic equation are called the roots of the equation .
Quadratic Equations In Standard Formax2 +bx + c = 0
ANDSince the function's value (y) is zero when ax2 + bx + c = 0, the solutions are also called zeros of the function f(x) = ax2 +bx + c.
NOTE:
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Use the "zero product property".
To solve ax2 +bx + c = 0:
If A B = 0, then A = 0 or B = 0
a. 3x - 6 = x2 - 10
1. Set = to 0 (may need to move terms).2. Factor.3. Set each factor = to 0.4. Solve for the variable.
b. Find the zeros of f(x) = 3x2 + 10x - 8.
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
c. What are the roots of the equationx2 - 5x - 36 = 0?
d. 3x2 + 4x = 4 e. 16x2 = 49
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
f. 3x2 +24x + 45 = 0 g. 10x2 = 9x
PracticeSolve by factoring.
1. 4x2 = 24x
2. 16x2 - 361 = 0
3. 20x = 25x2 + 4
4. 2x2 + 7x - 15 = 0
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
AnswersSolve by factoring.
1. 4x2 = 24x
2. 16x2 - 361 = 0
3. 20x = 25x2 + 4
4. 2x2 + 7x - 15 = 0
x = 0, 6
x = + 19/4
x = 2/5
x = -5, 3/2
Word Problems AGAIN!!
Doubling Area
5.3 Factoring and Solving Quadratics (work).notebook October 22, 2020
Extra ExampleYou have a rectangular vegetable garden in your backyard that measures 15 feet by 10 feet. You want to double the area of the garden by adding the same distance x to the length and width of the garden. Find the value of x and the new dimensions of the garden.