Solving Quadratic Equations and Inequalities

Quadratic 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 ax2+bx+cax^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 a0a\neq0:

  • ax2+bx+c<0ax^2+bx+c<0
  • ax2+bx+c>0ax^2+bx+c>0
  • ax2+bx+c0ax^2+bx+c\leq 0
  • ax2+bx+c0ax^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 ax2+bx+c<0ax^2+bx+c<0 are the xx-values where the graph of f(x)=ax2+bx+cf(x)=ax^2+bx+c is below the xx-axis. If the graph has 2 xx-intercepts and the curve opens up, the solutions are all of the values between the xx-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
Solve the quadratic inequality:
2x27x>32x^2-7x>-3
Step 1
Perform operations on both sides so that one side of the inequality is equal to zero.
2x27x>32x27x+3>0\begin{aligned}2x^2-7x&>-3\\2x^2-7x+3&>0\end{aligned}
Step 2
Determine the related function of the inequality:
f(x)=2x27x+3f(x)=2x^2-7x+3
Graph the related function using a graphing utility. Then use the graph to approximate the xx-intercepts.
The xx-intercepts are 0.5 and 3.
Step 3
Observe where the function rule meets the condition in the inequality.
2x27x+3>02x^2-7x+3>0
The function is greater than zero above the xx-axis. It does not include the xx-intercepts, which are the boundary points (greatest and least values) of the solution set.

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

Solution
The solutions are {xx<0.5 or x>3}\left \{ x|x<0.5\textrm{ or }x>3 \right \}.
Step-By-Step Example
Solve a Quadratic Inequality Algebraically
Solve the quadratic inequality:
x2+x+2<0x^2+x+2<0
Step 1
Determine the related function of the inequality:
f(x)=x2+x+2f(x)=x^2+x+2
Determine the related equation of the inequality by using the standard form:
ax2+bx+c=0x2+x+2=0\begin{aligned} ax^2+bx+c&=0\\x^2+x+2&=0\end{aligned}
The related equation shows that the value of a=1a=1, b=1b=1, and c=2c=2. Use the discriminant from the quadratic formula, b24acb^2-4ac, to determine the number of solutions of the related equation.
b24ac(1)24(1)(2)187\begin{gathered}b^2-4ac\\(1)^2-4(1)(2)\\1-8\\-7\end{gathered}
The discriminant is negative. So, the equation has no xx-intercepts.
Step 2
Determine the position of the graph of:
f(x)=x2+x+2f(x)=x^2+x+2
Since it has no xx-intercepts, it must be entirely above or entirely below the xx-axis. Test one point.
f(0)=02+0+2=2\begin{aligned}f(0)&=0^2+0+2\\&=2\end{aligned}
The point (0,2)(0,2) lies above the xx-axis. So, the graph must be entirely above the xx-axis.
Solution

The solutions of x2+x+2<0x^2+x+2<0 are the values of xx, where x2+x+2x^2+x+2 is less than zero. That means that the graph of f(x)=x2+x+2f(x)=x^2+x+2 lies below the xx-axis.

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