10.4
Solving Quadratic Equations
in Factored Form
Goal
Solve quadratic
equations in factored
form.
How deep is a crater?
In Exercises 50 and 51
you will solve a quadratic
equation to find the depth of
the Barringer Meteor Crater.
Key Words
• factored form
• zero-product property
A polynomial is in factored form if it is written as the product of two or more
factors. The polynomials in the following equations are written in factored form.
x(x 7) 0
(x 2)(x 5) 0
(x 1)(x 3)(x 8) 0
A value of x that makes any of the factors zero is a solution of the polynomial
equation. That these are the only solutions follows from the zero-product
property, stated below.
ZERO-PRODUCT PROPERTY
Let a and b be real numbers. If ab 0, then a 0 or b 0.
If the product of two factors is zero, then at least one of the factors
must be zero.
EXAMPLE
1
Using the Zero-Product Property
Solve the equation (x 2)(x 3) 0.
Student Help
Solution
STUDY TIP
The fact that the
solutions 2 and 3
in Example 1 are the
only solutions is a
consequence of the
zero-product property.
(x 2)(x 3) 0
Write original equation.
x20
Set each factor equal to 0.
or
x30
x2
ANSWER 䊳
x 3
Solve for x.
The solutions are 2 and 3. Check these in the original equation.
Solution by Factoring
Solve the equation and check the solutions.
1. (x 1)(x 3) 0
588
Chapter 10
Polynomials and Factoring
2. x (x 2) 0
3. (x 5)(x 7) 0