Solving Quadratic Equations by Factoring
Solve: 3x
2 - 9x = 120
Solution
| Step 1 Write the quadratic equation in
the form ax2 + 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.
|
3x2 - 9x = 120
3x2 - 9x - 120 = 0
3(x2 - 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 ax2 + 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 = x2 - 3x - 4
0 = x2 - 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 = ax2 + bx + c is a quadratic
equation.
