Solving Quadratic Equations by Factoring
Solve: 3x^{2}  9x = 120
Solution
Step 1 Write the quadratic equation in
the form ax^{2} + bx + c = 0.
Subtract 120 from both sides.
Step 2 Factor the polynomial.
Factor out the GCF, 3.
To factor the trinomial, find two integers
whose product is 40 and whose sum
is 3. They are 8 and 5.

3x^{2}  9x = 120
3x^{2}  9x  120 = 0
3(x^{2}  3x  40) = 0
3(x  8)(x + 5) = 0

Step 3 Use the Zero Product Property.
Set each binomial factor equal to 0.
Step 4 Solve each equation.
There are two solutions.
Step 5 Check each answer.
We leave the check to you. 
x  8 = 0 or x + 5 = 0
x = 8 or x = 5

Note:
When we used the Zero Product Property,
you may wonder why we did not set the
factor 3 equal to 0. Of course, 3 is not
equal to 0.
Furthermore, the product
3(x  8)(x + 5) is 0 because either (x  8) is 0 or (x + 5) is 0.
The constant 3 does not make the
product 0.
Example 2
Solve: 6 = (x  4)(x + 1)
Solution
Step 1 Write the quadratic equation in
the form ax^{2} + bx + c = 0.
Multiply the binomials on the right side. Then simplify.
Subtract 6 from both sides.
Step 2 Factor the polynomial.
Find two integers whose product is 10
and whose sum is 3. They are 5 and 2.

6 = (x  4)(x + 1)
6 = x^{2}  3x  4
0 = x^{2}  3x  10
0 = (x  5)(x + 2) 
Step 3 Use the Zero Product Property.
Set each factor equal to 0.
Step 4 Solve each equation.
Step 5 Check each answer.
We leave the check to you. 
x  5 = 0 or x + 2 = 0
x =5 or x = 2 
Note:
We can also write a quadratic equation
with 0 on the left side:
That is, 0 = ax^{2} + bx + c is a quadratic
equation.
