ALGEBRA 2:MODULE 4 LESSON 1

Solving Quadratic Equations by Factoring

What are the dimensions of the garden?

Q: Suppose you are trying to createa garden. The length of the gardenneeds to be six feet longer than the width. You will be given 40 squarefeet of space.

A: We know that the length is (x + 6) and the width is (x). The area is 40 sq. ft. If area equals length times width, then (x)(x+6) = 40. We want to find the values of that which make this equation true. We can use the method of factoring to solve this problem. Let’s learn how to solve by factoring so we can solve this problem without guess-and-check.

Image courtesy of Tom Curtis @ FreeDigitalPhotos.net

Distributive Property

Distributive Property a(b + c) = ab + ac Using the Distributive PropertyEX 1: 3ac(2a – 7c) = 6a²c – 21ac²EX 2: (4x – 5y)(4x + 5y) = 16x² - 25y²EX 3: (2x – 5)(x – 3) = 2x² - 11x + 15

Work for Example 2: Work for Example 3:

FOIL

FOIL

Methods for Factoring Quadratics

Greatest Common Factor- remove the greatest common factor from each term using the distributive property.

GCF (leftover)

Difference of Two Perfect Squares- factor into two binomials with opposite signs and the square root of each term. x² - y² = (x + y)(x – y)

Trinomial- factor into two binomials

x² + bx + c = (x + p)(x + q)

Greatest Common Factor

Factor 2x² + 4x = 2x(x + 2)

You should check each term to see what they ALL have in common and then factor it out of each term.

You can check by the distributive property.

Factor 90an²+18a²n – 9an = 9an(10n + 2a – 1)

GCF (leftover factor for each term)

GCF (leftover factor for each term)

Difference of Two Perfect Squares

EX 1: Factor 81x² - 16 = (9x) ² - 4² = (9x + 4)(9x – 4)

EX 2: Factor n² - 49 = (n) ² - 7² = (n + 7)(n – 7)

*Notice that EX 2 could be written as n² + 0n - 49 and when you multiply the binomials, the middle terms are eliminated.

*Notice that b = 0 and c = -49

*Let’s look at the X factor. Do you see the pattern?

This method does NOT apply to the SUM a² + b².

How to Factor a Trinomial, a = 1

Set up your two binomial factors (x + p)(x + q) for the quadratic x² + bx + c, if it is factorable.

Square root the first squared variable and put that in the front of each term

Set up the X factor with c at the top and b at the bottom

Find the values of p and q using the sum b and product c What 2 numbers multiply to = c AND add to = b? These are p and q!

Find this number p

Find this number q

Factor Trinomial Examples, a = 1

EX 1: Factor x2 + 8x + 7 = ( )( )-Write x in the front of each binomial-Find 2 numbers that multiply = 7 AND add = 8 *pq = 7 and p+q = 8-Put those numbers into the binomials

EX 2: Factor m2 – 2m – 80 = ( )( )-Write m in the front of each binomial-Find 2 numbers that multiply = -80 AND add = -2 *pq = -80 and p+q = -2-Put those numbers into the binomials

EX 3: Factor c2 + 4c - 12 = ( )( )-Write c in the front of each binomial-Find 2 numbers that multiply = -12 AND add = 4 *pq = -12 and p+q = 4

-Put those numbers into the binomials

Solutions for Factoring Trinomial, a = 1

EX 1: Factor x2 + 8x + 7 = (x + 7)(x + 1)-Write x in the front of each binomial-Find 2 numbers that multiply = 7 AND add = 8 *pq = 7 and p+q = 8 7 x 1 = 7 AND 7 + 1 = 8 Check them first!-Put those numbers into the binomials

EX 2: Factor m2 – 2m – 80 = (m - 10)(m + 8)-Write m in the front of each binomial-Find 2 numbers that multiply = -80 AND add = -2 *pq = -80 and p+q = -2 -10 x 8 = -80 AND -10 + 8 = -2 Check them first!-Put those numbers into the binomials

EX 3: Factor c2 + 4c - 12 = (c + 6)(c - 2)-Write c in the front of each binomial-Find 2 numbers that multiply = -12 AND add = 4 *pq = -12 and p+q = 4 6 x -2 = 4 AND 6 + -2 = 4 Check them first!

-Put those numbers into the binomials

**NOTE: The binomial factors may be reversed because multiplication is commutative!

7 1

-10 8

6 -2

How to Factor a Trinomial, a ≠1 Check to see if there is a GCF and factor it out You will have two binomial factors ( )( ) for the

quadratic ax² + bx + c, if it is factorable. Set up the X factor with the product ac at the top and b

at the bottom Find the values of p and q using the sum b and product ac *m x n = ac AND m + n = b Place m and n in the box with a and c terms Find the GCF of each row & column Use the signs from the m and n terms The GCFs are the binomial factors!

Find this number m

Find this number n

GCF

( )

ax² mx

nx cFind GCF

FindGCF

Binomial

(Bin

om

ial)

Factor the Trinomial, a ≠1

EX 1: Factor 2x2 + 13x - 7 =( )( )-Find ac and b, put them in the X factor

-Find m and n such that

mxn= -14 AND m+n = 13

14 x -1 = -14 AND 14 + -1 = 13

-Put them in box and find the GCF for each row and column.

2x2 + 13x - 7 =(x + 7)(2x - 1)

EX 2: Factor 6x2 - 19x + 3 =( )( )

-Find ac and b, put them in the X factor

-Find m and n such that

mxn= 18 AND m+n = -19

-18 x -1 = 18 AND -18 + -1 = -19

-Put them in the box and find the GCF for each row and column.

6x2 - 19x + 3 =(6x - 1)(x - 3)

GCF

x + 7

2x 2x2 14x

- 1 -1x - 7

GCF x - 3

6x 6x2 - 18x

- 1 - 1x 3

14 -1

-18 -1

Forms of a Quadratic Equation

Standard Form of a Quadratic Equation ax² + bx + c = 0, a ≠ 0

Factored Form of a Quadratic Equation a(x + p)(x + q) = 0, a ≠ 0

Factoring means to write the terms in multiplication form (as a product).

Zero Product Property

If ab = 0 then either a = 0 or b = 0 (or both).

The expression must be set equal to zero to use this property.

Zero Product Example: Quadratic in Factored Form (x – 6) (x + 8) = 0

x – 6 = 0 or x + 8 = 0

x = 6 or x = 8

Solve a Quadratic in Factored Form

Solve (x – 11)(x – 1) = 0.This equation is already in factored form. All we have to do is use the Zero Product Property and set each factor equal to zero.

x – 11 = 0   or   x – 1 = 0 x = 11  or  x = 1

You may write your solutions as x = {1, 11}. Do not enclose the values of x in parentheses. They may be misconstrued as an ordered pair! But brackets may be used to signify a set of numbers.

Solve by Factoring

1. Write the quadratic equation in standard form: move all terms to one side of the equation, usually the left, using addition or subtraction such that one side of the equation is equal to zero. Simplify if needed. *To make the process easier, lets always try to make our quadratic term positive! [ax² + bx + c = 0]

2. Write the quadratic equation in factored form: factor the equation completely. [a(x + p)(x + q) = 0]

3. Set each factor equal to zero, and solve each equation. [If ab = 0 then either a = 0 or b = 0]

Solve Quadratics by Factoring- Example 1

Solve x2 - 4x + 3 = 0. The quadratic equation is in standard form

(set equal to zero). This needs to be factored first.

Factor: x2 - 4x + 3 = (x - 1)(x - 3) *You may want to use the X factors for this.

(x - 1)(x - 3) = 0 Set each factor equal to zero: x - 1 = 0 or x - 3 = 0 Solve: x = 1  or  x = 3, can also write {1, 3}

Solve Quadratics by Factoring- Example 2

Solve x2 – 8 = 2x. Write the equation in standard form (set

the equation equal to zero). x2 – 2x – 8 = 0

Factor: x2 – 2x – 8 = (x – 4)(x + 2) (x – 4)(x + 2) = 0 Set each factor equal to zero: x – 4 = 0  or  x + 2 = 0 Solve: x = 4  or  x = -2 or {4, -2}

Solve Quadratics by Factoring- Example 3

Solve x² + 3x = 0 Since it is written in standard form, factor it using GCF: x(x + 3) = 0. Set all the factors equal to zero, and solve: x(x + 3) = 0

x = 0  or  x + 3 = 0 x = 0  or  x = –3 OR {0, -3}

Common mistakes on this type of problem: Do not use X factors- that is for Trinomials. Do not try to "solve" the equation for "x(x + 3) = 0" by dividing by x. Due

to the fact that the zero product property says either factor could be zero or both dividing the x assumes that x is not zero. We don’t know that, and you lose one of your solutions. Therefore, we can't divide by zero…it against math laws!

Even though you are used to factors having variables and numbers (like the factor, x + 3), a factor might contain only a variable, such as x. Be sure to set this equal to zero also. If it said 5(x-2)(x+4) = 0, 5 cannot = 0!

Solve Quadratics by Factoring- Example 4

Solve (x + 2)(x + 3) = 20.

It is very common to see this type of problem, and say:It's already factored!  So I'll set the factors equal to 0 and solve.Remember, the expression must be set equal to zero to use this property.

Simplify by the distributive property (x + 2)(x + 3) = 20 (x + 2)(x + 3) -20 = 0

x2 + 5x + 6 - 20 = 0x2 + 5x – 14 = 0 It is in standard form, ready to factor(x + 7)(x – 2) = 0 Factored, can use X factorsx + 7 = 0  or   x – 2 = 0 Set each factor = to 0 and solvex = –7  or   x = 2

Solve Quadratics by Factoring- Example 5

Solve 8x2 = 26x - 15. Write the equation in standard form (set the

equation equal to zero). 8x2 - 26x + 15 = 0 Factor: (4x – 3)(2x - 5) = 0 Set each factor equal to zero: 4x – 3 = 0  or  2x - 5 = 0 Solve: 4x – 3 = 0 or  2x - 5 = 0

4x = 3 or 2x = 5x = ¾ or x = 5/2

-20 -6

2x - 5

4x 8x2 -20x

- 3 -6x 15

Back to the Garden

Q: Suppose you are trying to createa garden. The length of the gardenneeds to be six feet longer than the width. You will be given 40 squarefeet of space.

A: We know that the length is (x + 6) and the width is (x). The area is 40 sq. ft. If area equals length times width, then (x)(x+6) = 40. We want to find the values of that make this equation true. We can use the method of factoring to solve this problem. Simplify using the distributive property: x² + 6x = 40Write it in standard form: x² + 6x – 40 = 0Factor: (x + 10) (x – 4) = 0Set each factor equal to zero:x + 10 = 0 or x – 4 = 0Solve: x = -10 or x = 4 Since length cannot be negative, -10 cannot be a solution and the only solution is width = 4. Use 4 to find the length 4 + 6 = 10. The dimensions will be 4 feet x 10 feet.

Image courtesy of Tom Curtis @ FreeDigitalPhotos.net