Solving a Quadratic Inequality
A quadratic inequality is a lot like a quadratic equation, except that the equals sign is
replaced by less than (<), greater than (>), less than or equal to () or greater than or
equal to (). For example:
4 Â· x^{2}  2
Â· x + 3
19.
When you solve a quadratic inequality, the idea is to rearrange the inequality to make x
the subject. Unlike solving an equation, when you solve a quadratic inequality, you can
get one interval of xvalues as your solution, or you can get two intervals of xvalues as
your solution.
Preliminary Steps in Solving a Quadratic Inequality
When the coefficient of x^{2} (i.e. the number represented by a) is positive, quadratic
inequalities are much easier to solve if they are in one of the four forms:
a Â· x^{2} + b
Â· x + c > 0
a Â· x^{2} + b
Â· x + c < 0
a Â· x^{2} + b
Â· x + c
0
a Â· x^{2} + b
Â· x + c
0
where b and c are numbers that might be positive, negative or zero. The truly important
things about these forms are:
Â· One side of the inequality has been reduced to zero.
Before you begin solving the quadratic inequality, it is a good idea to manipulate the
inequality so that that it is one of the four forms listed above.
