Solving Quadratic Equations
Solve a quadratic equation by factoring
Find the zeros of a quadratic function
The factoring techniques you have learned provide us with tools for solving equations that
can be written in the form
This is a quadratic equation in one variable, here x.
You can recognize such a quadratic equation by
the fact that the highest power of the variable x is
the second power.
in which a, b, and c are constants.
An equation written in the form ax
0 is called a quadratic equation in
standard form. Using factoring to solve quadratic equations requires the zero-product
rule, which says that if the product of two factors is 0, then one or both of the factors must
be equal to 0. In symbols:
Rules and Properties: Zero-Product Rule
If a b
0, then a
0 or b
0 or a
Let’s see how the rule is applied to solving quadratic equations.
Solving Equations by Factoring
Factoring on the left, we have
To use the zero-product
rule, 0 must be on one side of
By the zero-product rule, we know that one or both of the factors must be zero. We can then
Solving each equation gives
Graph the function
on your graphing calculator.
The solutions to the equation
The two solutions are 6 and
3 and the solution set is written as x x
3, 6 or simply
18 will be the x
3, 6 .
coordinates of the points on
The solutions are sometimes called the zeros, or roots, of the equation. They represent
the curve at which y
the x coordinates of the points where the graph of the equation y
are the points at which the
the x axis. In this case, the graph crosses the x axis at ( 3, 0) and (6, 0).
graph intercepts the x axis.