1. 5.2 Solving Quadratic
Equations by Factoring

2. What is Factoring?
 Used to write trinomials as a product of
binomials.
 Works just like FOIL in reverse.
 Example: Multiply (x + 2)(x + 7)
 What do you notice about the 2 and 7?
 2 + 7 = 9 and 2 x 7 = 14
 In general: (x + m)(x + n) = ax2 + bx + c
◦ where a=1, b = m + n, and c = mn

3. Factoring x2 + bx + c
 (x + m)(x + n) = x2 + bx + c
 Need to find m and n so m+n = b and
mn = c
 First, find all factor pairs of c.
 Find their sums. Choose the pair whose
sum equals b.
 Example: Factor x2 + 5x + 6
Factor Pairs Sum

4. Example:
 What if b is negative and c is positive?
◦ Choose negative factors!
x2 - 7x + 10

5. Example:
 What if c is negative?
◦ Choose one positive and one negative!
x2 - 8x – 20

6. Your Turn!
 Factor x2 - 2x – 48

7. Factoring ax2 + bx + c (a  1)
 Need k, l, m and n, such that:
ax2 + bx + c = (kx +m)(lx + n)
 So, kl = a and mn = c.
 Find factors of a and c, then check
possible answers.
 Example: Factor 3x2 - 17x + 10

8. Example:
 Factor 4x2 - 4x – 3

9. Your Turn!
 Factor 3x2 + x – 10

10. Factoring Special Patterns
Difference of Two Squares:
a2 – b2 = (a + b)(a – b)
Example: Factor x2 – 9
Perfect Square Trinomial:
a2 + 2ab + b2 = (a + b) 2
a2 - 2ab + b2 = (a - b) 2
Example: Factor x2 + 12x + 36

11. Examples:
 Factor 4x2 – 25
 Factor 9x2 + 24x +16
 Factor 49x2 – 14x + 1

12. Factoring Out Monomials
 Check if you can factor out something
from each term.
 Example: Factor 5x2 – 20
 Example: Factor 2x2 + 8x

13. Your Turn!
 Factor 6x2 + 15x + 9

14. Solving Quadratics by Factoring
 Certain quadratic equations can be
solved by factoring.
 Standard form: ax2 + bx + c = 0
 Zero Product Property:
◦ Let A and B be real numbers or algebraic
expressions. If AB = 0, then A = 0, or B = 0.

15. Example:
 Solve x2 + 3x – 18 = 0

16. Finding Zeros
 The x-intercepts of a function are also
called zeros.
 To find zeros:
◦ Factor to rewrite in intercept form.
◦ y = ax2 + bx + c  y = a(x – p)(x – q)
◦ p and q are the zeros

17. Example:
 Find the zeros of y = x2 – x – 6