1. 10.4 Solving Equations in Factored Form
Algebra
Solving Equations in Factored Form

2. Quadratic Equations in Standard Form
You have already learned the standard form of a quadratic equation:
ax² + bx + c = 0 where a ≠ 0

3. Quadratic Equations in Factored Form
Consider the equation: x² + 8x + 15 = 0
Here is the same equation, in factored form:
(x + 5)(x + 3) = 0
FOIL to be sure x²
+3x
+5x
+15
x² + 8x + 15 = 0

4. Two Ways to Solve: x² + 8x + 15 = 0
Quadratic Formula
Factored form
(x + 5)(x + 3) = 0
-8 ± 8² - 4(1)(15) √
x =
2(1)
This means that for the equation to be true, either:
-8 ± √ 4
x = or
x + 5 = 0
x + 3 = 0 2
-8 ± 2 -6
-10
x = =
x = -5
x = -3 2 2 2
x = -3, -5
x = -3, -5

5. ZERO PRODUCT PROPERTY
If a product equals zero, then one of the factors must be 0.
If ab = 0, then either
a = 0
b = 0
or both = 0

6. Apply the ZERO PRODUCT PROPERTY……
We said that the solutions to the equation
x² + 8x + 15 = 0 or (x +5)(x + 3) are and -5.
Substitute each solution into the equation in factored form:
(-3 + 5)(-3 + 3) = 0
Substitute -3 for x
(2) (0) = 0 TRUE
(-5 + 5)(-5 + 3) = 0
Substitute -5 for x
(0) (-2) = 0 TRUE
Substitute into original equation:
(-3)² + 8(-3) + 15 = 0
True
(-5)² + 8(-5) + 15 = 0
True

7. Try these (x + 7) (x – 3) = 0 x = -7 or x = 3 (x - 4)(x – 5) = 0
+5 +5
+3 +3
2x = 5
x = 3
2 2 5
x =
x = 3 2

8. Shortcut: (x + 7) (x – 3) = 0 -7 +3
When the factors are in the form (x +__ ) or (x - __ ) the solutions will always be the opposite of the factors, in this case -7 and +3.
(x + 7) (x – 3) = 0
Ask yourself: what value of x will make this binomial = 0?
Ask yourself: what value of x will make this binomial = 0? -7 +3

9. Shortcut: (3 + 2) (3x + 2) (x – 1) = 0 The other solution is +1 -2 = 0
Ask yourself: what value of x will make this binomial = 0?
The other
solution is +1 -2
( )
= 0 3
The denominator would cancel with 3 when multiplied
The numerator would cancel with +2 when added 2
One solution is - 3

10. Try these x = -½ or x = 3 (2x + 1) (x – 3) = 0 (3x - 4)(2x + 5) = 0

11. A Few together from the HW
P. 600 #29, 43

12. Homework
pg # ,