## Solving Quadratic Equations and Inequalities

A quadratic inequality may have zero, one, or infinitely many real solutions. The solution set can be identified by graphing the related function and determining which intervals of the domain satisfy the inequality.

A quadratic inequality is a mathematical statement that compares a quadratic expression of the form $ax^2+bx+c$ to another expression or value using the symbols $<$, $>$, $\leq$, or $\geq$. A quadratic inequality can be expressed in one of these forms, where $a\neq0$:

• $ax^2+bx+c<0$
• $ax^2+bx+c>0$
• $ax^2+bx+c\leq 0$
• $ax^2+bx+c\geq 0$

A solution of an inequality is any value of the variable that makes the inequality true, which means the comparison is valid. A quadratic inequality may have zero, one, or infinitely many real solutions. When there are infinitely many solutions, they may be represented on a number line or by using set builder notation (a shorthand way of writing a set of values).

The number of solutions of a quadratic inequality can be determined by looking at the graph. For example, the solutions of $ax^2+bx+c<0$ are the $x$-values where the graph of $f(x)=ax^2+bx+c$ is below the $x$-axis. If the graph has 2 $x$-intercepts and the curve opens up, the solutions are all of the values between the $x$-intercepts, so there are infinitely many solutions.

To solve a quadratic inequality, the solution set may be found by graphing the related function and determining which intervals of the domain satisfy the inequality.

Step-By-Step Example
Graph to Solve a Quadratic Inequality
$2x^2-7x>-3$
Step 1
Perform operations on both sides so that one side of the inequality is equal to zero.
\begin{aligned}2x^2-7x&>-3\\2x^2-7x+3&>0\end{aligned}
Step 2
Determine the related function of the inequality:
$f(x)=2x^2-7x+3$
Graph the related function using a graphing utility. Then use the graph to approximate the $x$-intercepts.
The $x$-intercepts are 0.5 and 3.
Step 3
Observe where the function rule meets the condition in the inequality.
$2x^2-7x+3>0$
The function is greater than zero above the $x$-axis. It does not include the $x$-intercepts, which are the boundary points (greatest and least values) of the solution set.

The function is positive, or greater than zero, when $x<0.5$ or when $x>3$.

Solution
The solutions are $\left \{ x|x<0.5\textrm{ or }x>3 \right \}$.
Step-By-Step Example
$x^2+x+2<0$
Step 1
Determine the related function of the inequality:
$f(x)=x^2+x+2$
Determine the related equation of the inequality by using the standard form:
\begin{aligned} ax^2+bx+c&=0\\x^2+x+2&=0\end{aligned}
The related equation shows that the value of $a=1$, $b=1$, and $c=2$. Use the discriminant from the quadratic formula, $b^2-4ac$, to determine the number of solutions of the related equation.
$\begin{gathered}b^2-4ac\\(1)^2-4(1)(2)\\1-8\\-7\end{gathered}$
The discriminant is negative. So, the equation has no $x$-intercepts.
Step 2
Determine the position of the graph of:
$f(x)=x^2+x+2$
Since it has no $x$-intercepts, it must be entirely above or entirely below the $x$-axis. Test one point.
\begin{aligned}f(0)&=0^2+0+2\\&=2\end{aligned}
The point $(0,2)$ lies above the $x$-axis. So, the graph must be entirely above the $x$-axis.
Solution

The solutions of $x^2+x+2<0$ are the values of $x$, where $x^2+x+2$ is less than zero. That means that the graph of $f(x)=x^2+x+2$ lies below the $x$-axis.

Since the graph is entirely above the $x$-axis, the inequality has no solutions.