1. (touches x-axis twice) (does not touch x-axis)
Section 10-5 Factor to Solve Quadratic Equations SPI 23E: Find the solution to a quadratic equation given in standard form SPI 23F: select the solution to a quadratic equation given solutions represented in graphical form
Objectives:
Solve (find real number solutions) by factoring
The solutions of a quadratic equation are:
the x-intercepts since they are on the real number line
a quadratic equation can have solutions as follows:
Two Solutions
(touches x-axis twice)
One Solutions
(touches x-axis once)
No Solution
(does not touch x-axis)

2. Methods of Finding Real Number Solutions
Two Methods for Solving Quadratic Equations:
ax + by = c
Factoring and using the Zero-Product Property
Using the Quadratic Formula

3. Zero-Product Property
Factor x2 + 7x + 10 = 0
Factors of 10
Sum of Factors
1 ∙ 10 11
2 ∙ 5 7
Both conditions must be true
Factors of the problem are (x + 2)(x + 5)
To find the real number solutions, use the
Zero-Product Property
Solve (x + 2)(x + 5) = 0
x + 2 = or x + 5 = 0
x + 2 – 2 = 0 – or x + 5 – 5 = 0 – 5
x = or x = - 5
There are two real number solutions to the quadratic equation.

4. Factor to Solve a Quadratic Equation
Try It!
Factor to Solve a Quadratic Equation
Solve x2 + x – 42 = 0 by factoring.
Factors of - 42
Sum of Factors
2 ∙ -21
-19
-6 ∙ 7
= 1
Both conditions must be true
Factors of the problem are (x + 7)(x -6)
(x + 7)(x – 6) = 0
Factor using x2 + x – 42
x + 7 = 0 or x – 6 = 0
Use the Zero-Product Property.
x = –7 or x = 6
Solve for x.

5. Practice
1. Use the zero property to solve (x – 3) (x – 7) = 0.
3 and 7
2. Use the zero property to solve – 3n(2n – 5) = 0.
0 and 2.5
3. Solve by factoring x2 – 16 x + 55 = 0.
5 and 11
4. Solve by factoring m2 – 5m – 14 = 0.
-2 and 7