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Algebra 1

8.9.2 Using Factored Form and the Zero Product Property to Solve Quadratic Equations

Algebra 18.9.2 Using Factored Form and the Zero Product Property to Solve Quadratic Equations

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Activity

To solve the equation n22n=99n22n=99, Tyler wrote out the following steps. Analyze Tyler's work by explaining what Tyler did in each step.

Original equation n22n=99n22n=99

Step 1 - n22n99=0n22n99=0

1.

What did Tyler do in Step 1?

Step 2 - (n11)(n+9)=0(n11)(n+9)=0

2.

What did Tyler do in Step 2?

Step 3 - n11=0n11=0 or n+9=0n+9=0

3.

What did Tyler do in Step 3?

Step 4 - n=11n=11 and n=9n=9

4.

What did Tyler do in Step 4?

For questions 5 – 9, solve each equation by rewriting it in factored form and using the zero product property.

5.

x 2 + 8 x + 15 = 0 x 2 + 8 x + 15 = 0

6.

x 2 8 x + 12 = 5 x 2 8 x + 12 = 5

7.

x 2 10 x 11 = 0 x 2 10 x 11 = 0

8.

49 x 2 = 0 49 x 2 = 0

9.

( x + 4 ) ( x + 5 ) 30 = 0 ( x + 4 ) ( x + 5 ) 30 = 0

Are you ready for more?

Extending Your Thinking

1.

Solve this equation and explain or show your reasoning.

( x 2 x 20 ) ( x 2 + 2 x 3 ) = ( x 2 + 2 x 8 ) ( x 2 8 x + 15 ) ( x 2 x 20 ) ( x 2 + 2 x 3 ) = ( x 2 + 2 x 8 ) ( x 2 8 x + 15 )

Self Check

Solve the equation by rewriting it in factored form and then using the zero product property.

x 2 + 3 x 59 = 5

  1. x = 9 and x = 6
  2. x = 9 and x = 6
  3. x = 9 and x = 6
  4. x = 9 and x = 6

Additional Resources

Using Factored Form and the Zero Product Property to Solve Quadratic Equations

We have seen quadratic equations in many different forms. Let’s look at an example in which we find the factored form and then use the zero product property to solve.

Example 1

Solve x2+3x50=10x2+3x50=10.

Step 1 - Set the equation equal to 0 by adding 10 to both sides.

x2+3x40=0x2+3x40=0

Step 2 - Find the factored form.

Which numbers have a product of –40 and a sum of +3?

(x+8)(x5)=0(x+8)(x5)=0

Step 3 - Use the zero product property to solve.

Set each factor equal to 0 and solve.

(x+8)=0(x5)=0(x+8)=0(x5)=0

x=8x=5x=8x=5

The solution is x=8x=8 and x=5x=5.

Example 2

Solve x2+12x+21=15x2+12x+21=15.

Step 1 - Set the equation equal to 0 by adding 15 to both sides.

x2+12x+36=0x2+12x+36=0

Step 2 - Find the factored form.

Which numbers have a product of 36 and a sum of 12?

(x+6)(x+6)=0(x+6)(x+6)=0

Step 3 - Use the zero product property to solve.

Set each factor equal to 0 and solve. The factors are the same, so there is one solution.

(x+6)=0(x+6)=0(x+6)=0(x+6)=0

x=6x=6x=6x=6

The only solution is x=6x=6.

Try it

Try It: Using Factored Form and the Zero Product Property to Solve Quadratic Equations

1.

Solve z25z48=12z25z48=12.

2.

Solve p23p108=20p23p108=20.

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