Solving by factoring

SOLVING BY FACTORING

I can use the zero product property to solve quadratics by factoring

Warm UpUse your calculator to find the x-intercept of each function.

1. f(x) = x2 - 6x + 8 2. f(x) = -x2 – 2x + 3

Factor each expression.

3. 3x2 – 12x 4. x2 – 9x + 18

5. x2 – 49

3x(x – 4)

(x –7)(x +7)

(x –6)(x –3)

Connections

We find zeros on a graph by looking at the x-intercepts or viewing the table and identifying the x-intercept as the point where y=0.

Using this knowledge determine how one could find the zeros of a quadratic algebraically. Share your method with your partner.

Use f(x) = x2 – 3x – 18 to help your discussion.

You can find the roots of some quadratic equations by factoring and applying the Zero Product Property.

• Functions have zeros or x-intercepts.

• Equations have solutions or roots.

Reading Math

Find the zeros of the function by factoring.

Example 2A: Finding Zeros by Factoring

f(x) = x2 – 4x – 12

x2 – 4x – 12 = 0

(x + 2)(x – 6) = 0

x + 2 = 0 or x – 6 = 0

x= –2 or x = 6

Set the function equal to 0.

Factor: Find factors of –12 that add to –4.

Apply the Zero Product Property.

Solve each equation.

Find the zeros of the function by factoring.

Example 2B: Finding Zeros by Factoring

g(x) = 3x2 + 18x

3x2 + 18x = 0

3x(x+6) = 0

3x = 0 or x + 6 = 0

x = 0 or x = –6

Set the function to equal to 0.

Factor: The GCF is 3x.

Apply the Zero Product Property.

Solve each equation.

Check It Out! Example 2a

f(x)= x2 – 5x – 6

Find the zeros of the function by factoring.

A. B. g(x) = x2 – 8x

Quadratic expressions can have one, two or three terms, such as –16t2, –16t2 + 25t, or –16t2 + 25t + 2. Quadratic expressions with two terms are binomials. Quadratic expressions with three terms are trinomials. Some quadratic expressions with perfect squares have special factoring rules.

Find the roots of the equation by factoring.

Example 4B: Find Roots by Using Special Factors

18x2 = 48x – 32

Rewrite in standard form.

Factor. The GCF is 2.

Divide both sides by 2.

Write the left side as a2 – 2ab +b2.

Apply the Zero Product Property.

Solve each equation.

18x2 – 48x + 32 = 0

2(9x2 – 24x + 16) = 0

(3x – 4)2 = 0

3x – 4 = 0 or 3x – 4 = 0

x = or x =

9x2 – 24x + 16 = 0

(3x)2 – 2(3x)(4) + (4)2 = 0

Factor the perfect-square trinomial.

Find the roots of the equation by factoring.

Example 4A: Find Roots by Using Special Factors

4x2 = 25

Rewrite in standard form.

Factor the difference of squares.

Write the left side as a2 – b2.

Apply the Zero Product Property.

Solve each equation.

4x2 – 25 = 0

(2x)2 – (5)2 = 0

(2x + 5)(2x – 5) = 0

2x + 5 = 0 or 2x – 5 = 0

x = – or x =

A. x2 – 4x = –4

Find the roots of the equation by factoring.

Check It Out! Example 4a

B. 25x2 = 9

Write a quadratic function in standard form with zeros 4/3 and –7. Your factors should not include fractions.

Example 5: Using Zeros to Write Function Rules

Check It Out! Example 5

Write a quadratic function in standard form with zeros 5/2 and –5. Your factors should not include fractions.

Could you develop more than one quadratic with the same zeros?

If yes give an example use the zeros 2 and 4.

If no explain why.