Solving Quadratic Equations by Factoring
Example 1
Solve by factoring: x^{2}  14 = 5x
Solution
Step 1 Write the equation in the form ax^{2}
+ bx + c = 0 Add 5x to both sides.
Step 2 Factor the polynomial.
The coefficient of x^{2} is 1, so find two
numbers whose product is 14 and
whose sum is 5.

x^{2} + 5x  14 
= 0 
These numbers are 2 and 7.
Step 3 Use the Zero Product Property.
Step 4 Solve the resulting equations.
Step 5 Check each answer.
We leave the check to you. 
(x  2)(x + 7) x  2 = 0
x = 2 
= 0 or x + 7 = 0
or x = 7 
So, the two solutions of x^{2}  14 = 5x are x = 2 and x = 7.Example 2
Solve by factoring: 7x + 6x^{2} = 20
Solution
Step 1 Write the equation in the form ax^{2}
+ bx + c = 0 Subtact 20 from both sides.
Step 2 Factor the polynomial.
The coefficient of x^{2} is not 1, so find two numbers
whose product is ac and whose sum is b.
ac = 6(20) = 120 , b = 7
The two numbers are 8 and 15. 
6x^{2} + 7x  20 
= 0 
Rewrite the middle term,
7x, as 8x + 15x.
Factor 2x from the first pair
of terms.
Factor 5 from the second pair
of terms.
Factor out the common
binomial factor, 3x  4. Step 3 Use the Zero Product Property.
Step 4 Solve the resulting equations.
Step 5 Check each answer.
We leave the check to you. 
6x^{2}  8x + 15x
 20
2x(3x  4) + 5(3x  4)
(3x  4)(2x + 5) 3x  4 = 0 or 2x + 5

= 0
= 0
= 0 = 0

So, the two solutions of 7x + 6x^{2} = 20 are
