What is the solution to a Quadratic Inequalities. Interactive tutorial with pictures and examples

Before learning how to solve the equation of a quadratic inequality. It is important that you:

Can graph parabolas
Understand the vertex and standard form equation of a parabola
Solve quadratic equations

Quick review of the graphs and equations of linear inequalities.

The graphs of quadratic inequalities follow the same general relationship. Greater than inequalities are the region above the equation's graph and less than inequalities are made up by the region underneath the graph of the equation.

What is the solution of a quadratic inequality?

Below, you will learn a formula for solving quadratic inequalities. First, it's important to try to understand what a quadratic inequality is and what its solution is. So let us explore a graphical solution for a quadratic inequality.

We will examine the quadratic inequality $$ y > x^2 -1 $$ .

The yellow region represents the graph of the quadratic inequality.

The red line segment from $$ (-1, 2) $$ to $$ (1, 2) $$ represents the solution itself, graphically.

The solution, graphically, is always where the graph of the inequality overlaps with the x axis .

Diagram 7

The same basic concepts apply to quadratic inequalities like $$ y < x^2 -1 $$ from digram 8. This is the same quadratic equation, but the inequality has been changed to $$ \red < $$.

In this case, we have drawn the graph of inequality using a pink color. And that represents the graph of the inequality.

The solution , graphically, is always where the graph of the inequality overlaps with the x axis .

Diagram 8

General Formula
for Solutions of Quadratic Inequalities

The table below represents two general formulas that express the solution of a quadratic inequality of a parabola that opens upwards (ie a > 0) whose roots are r₁ and r₂.

The Greater Than Inequality

0 > ax² + bx + c

Solution: {x| r₁ < X < r₂}

The Less Than Inequality

0 < ax² + bx + c

Solution: {x| x < r₁ or x > r₂}

We can reproduce these general formula for inequalities that include the quadratic itself (ie ≥ and ≤).

The Greater Than Inequality

0 ≥ ax² + bx + c

Solution: {x| r₁ ≤ X ≤ r₂}

The Less Than Inequality

0 ≤ ax² + bx + c

Solution: {x| x ≤ r₁ or x ≥ r₂}

Warning about imaginary solutions:
Although the solution of a quadratic equation could be imaginary. The solution of a quadratic inequality cannot include imaginary numbers -- this is because imaginary numbers cannot be ordered.