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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.