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.


Examples: a. x2 + 6x + 8 b. 3x2 - 11x + 6

Examples: c. x2 + 7x - 18 d. 3x2 +10x - 8


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


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 =


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)


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


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)


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


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)


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:


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.


c. What are the roots of the equationx2 - 5x - 36 = 0?

d. 3x2 + 4x = 4 e. 16x2 = 49


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


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


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.

5.3 factoring and solving quadratics (work).notebook

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