Solving Quadratic Equations by Factoring
To solve certain quadratic equations, we will use the Zero Product
Property. This property states that if the product of two numbers
(or polynomials) is 0, then one (or both) factors is 0.
Property —
Zero Product Property
If P · Q = 0, then P = 0 or Q = 0 (or both P and Q
= 0).
Here, P and Q are polynomials.
In the next two examples, we use the Zero Product Property.
If w(w - 7) = 0, then w = 0 or w - 7 = 0.
If (y - 4)(y + 6) = 0, then y - 4 = 0 or y + 6 ) 0.
The following procedure can be used to solve a quadratic equation in
standard form where the second-degree polynomial is factorable over the
integers.
Procedure — To Solve a Quadratic Equation by Factoring
Step 1 Write the equation in the form ax2 + bx + c = 0.
Step 2 Factor the polynomial.
Step 3 Use the Zero Product Property.
Step 4 Solve the resulting equations.
Step 5 Check each answer.
Because the degree of a quadratic equation is 2, each quadratic equation
has two solutions.
Example 1
Solve by factoring: 6x2 = 8x
Solution
| Step 1 Write the equation in the form ax2
+ bx + c = 0. Subtract 8x from both sides.
Step 2 Factor the polynomial.
Factor out 2x.
Step 3 Use the Zero Product Property.
Set each linear factor equal to 0.
Step 4 Solve the resulting equations.
Divide both sides of the first equation by 2.
Add 4 to both sides of the second equation.
|
6x2 - 8x
2x(3x - 4)
2x = 0
x = 0 |
= 0
= 0
or 3x - 4 = 0
or 3x = 4 |
| Divide both sides of the second equation by 3. |
x = 0 |
or
 |
| Step 5 Check each answer. |
|
|
So, the quadratic equation 6x2
= 8x has two solutions, 0 and
