Warm-Up

Factor the following completely:Answers:

1.

2.

3.

4.

2 5 14x x 22 14 24x x

( 7)( 2)x x

2( 3)( 4)x x 2 10 24x x ( 12)( 2)x x 23 27 60x x 3( 4)( 5)x x

Lesson 13.7 Solving Equations by Factoring

Objectives:1. Learn and understand what the Zero Product Property

means.2. Use the Zero Product Property to solve quadratic equations

that have already been factored.3. Understand that a quadratic equation must be equal to zero

before factoring it.4. Solve an equation by factoring it.

Have you realized?

That every question we have reviewed so far has not had an “=“ sign.

All questions so far have been FACTOR, SIMPLIFY, or DISTRIBUTE.

Today’s lesson we will include SOLVE BY FACTORING into the mix.

The Zero Product Property Review from 7th grade states:

If the product of two or more factors is zero, then at least one of the factors must be zero.

Or in other words, if the problem equals zero, at least on of the “parts” has to be a zero.

Examples:4x = 0zy = 0

6wk = 0

x = 0z = 0 or y = 0w = 0 or k = 0

(x + #)(x + #) = 0The following is a Quadratic Equation that is in

factored form:

Using the Zero Product Property we know that if at least one set of parentheses (one set of bubbles)

equals a zero, then the entire problem will “zero out.”

(x + a) is one factor (x + b) is another factor

7 1 0x x Example: What number can I plug into each bubble to make it equal zero?

x = 7 and x = -1

Technically you take whatever is inside the bubble and set it equal to zero, like a mini equation and solve.

Examples:(x + 4)(x – 3) = 0x + 4 = 0 x – 3 = 0OR

(3z + 6)(z + 1) = 0y(2y – 8) = 0

x = -4 x = 3OR

3z + 6 = 0 z + 1 = 0OR

x = -2 x = -1OR

y = 0 2y – 8 = 0OR

y = 0 y = 4OR

3z = -6

2y = 8

1)

3)

2)

x = -4, 3z = -2, -1y = 0, 4

Now it’s your turn!(4x – 4)(2x – 3) = 04)

6)

5) 3t(t + 7) = 0

11(y – 2)(9x + 18) = 0

Now that we know the zero product property and how to use it, let’s apply it and solve some quadratic equations! (Keep in mind

this is going to require us to rearrange, factor, and solve)

This is gonna be awesome!

To solve a Quadratic Equation by factoring, it must first be equal to zero:ax2 + bx + c = 0Examples:x2 + 3x + 2 = 0

3k2 – 3k – 8 = -23k2 – 3k – 6 = 0+ 2+ 2

2n2 + 3n + 1 = -3n – 3 + 3+ 3+ 3n + 3n

2n2 + 3n + 4 = -3n2n2 + 6n + 4 = 0

1)

2)

3)

Once the Quadratic Equation is equal to zero the next step to solving it is by factoring:2n2 + 6n + 4 = 0Example: 2 (n2 + 3n + 2) = 02 (n + 1)(n + 2) = 0

After factoring, use the Zero Product Property to finish solving the quadratic equation:

(n + 1)(n + 2) = 0n + 1 = 0 n + 2 = 0

STEP 1: Factor out the GCF

STEP 2: Factor the Quadratic Equation

STEP 3: Divide both sides by 2

OR STEP 4: Solve both equations

n = -1, -2 STEP 5: Check answer by substituting back into original equation

(n + 1)(n + 2) = 0

Let’s do a recap of the steps…(this might be something you want to write down for future reference!)

Step 1: Make sure the entire equation is set = to 0. (If it is not, you better do some rearranging to get all letters and numbers on one side!)

Step 2: Pull out a Greatest Common Factor. (If you can) (In other words, pull out a number or variable that divides all terms evenly!)

Step 3: Factor the leftover quadratic into two binomials. (Meaning you should have 2 sets of parentheses) ex. (x + 2)(x – 3)

Step 4: Use the Zero Product Property to set each binomial equal to 0.

Step 5: Solve each binomial for the variable.

Step 6: “GLADE” IT! (“Plug it in, plug it in”) to check your work and see if your answers make sense.

S’More Practice!

2n2 – 8n + 8 = 06a2 + 15a – 14 = -5

5b2 – 80 = 0

1) 2)

3) 4) 2x2 + 17x + 25 = x – 5

And still s’more practice!

2n2 + 8n – 5 = -52a2 – 16a + 34 = 4

3b2 – 18b + 27 = 0

5) 6)

7) 8) 6x2 + 12x – 29 = 3x – 2

Homework:

Complete the 13.7

(And bring in smores to your math teacher!)