Quadratic Inequality Calculator (Free, Step-by-Step)

Use this free quadratic inequality calculator to solve inequalities like ax² + bx + c > 0. It finds the roots, determines where the inequality holds, and gives the solution in interval notation with a shaded graph.

How to solve a quadratic inequality

Find the roots of the related equation, then test the sign of the parabola on each interval. An upward parabola is positive outside its roots and negative between them (and vice versa).

How to use it

1. Enter a, b, c and pick the inequality sign.
2. Press Solve.

Related: the Inequality Calculator and more free math calculators.

Frequently asked questions

What is a quadratic inequality?

A quadratic inequality compares a quadratic expression to a value using <, >, ≤ or ≥, such as x² − 5x + 6 > 0. Instead of one or two answers, the solution is usually a range of x values. Here x² − 5x + 6 factors as (x − 2)(x − 3), and the inequality is true for x < 2 or x > 3.

How do you solve quadratic inequalities?

Move everything to one side so the other side is 0, then solve the related equation to find the roots. The roots split the number line into intervals. Test one value from each interval in the original inequality. For x² − 4 ≤ 0, the roots are −2 and 2, and testing x = 0 gives −4 ≤ 0, so the solution is [−2, 2].

What is the difference between a quadratic equation and a quadratic inequality?

A quadratic equation sets the expression equal to a value and has at most two real solutions. A quadratic inequality uses <, >, ≤ or ≥ and usually has a whole interval of solutions. For example, x² = 9 has only x = −3 and x = 3, while x² < 9 is true for every x between them, written (−3, 3).

How do you solve quadratic inequalities with a graphing calculator?

Graph y equal to the quadratic expression and find where the curve crosses the x-axis. For > 0, the solution is where the graph sits above the axis; for < 0, it is where the graph sits below. With y = x² − 2x − 8, the zeros are −2 and 4, so x² − 2x − 8 < 0 holds on the interval (−2, 4).

How do you write a quadratic inequality solution in interval notation?

Use parentheses for strict signs (< or >) and square brackets for ≤ or ≥, since those include the roots. Join separate pieces with the union symbol ∪. For example, x² − 1 ≥ 0 is true when x ≤ −1 or x ≥ 1, written (−∞, −1] ∪ [1, ∞). Infinity always gets a parenthesis because it is not a value you can reach.

How to use the Quadratic Inequality Calculator for homework

The Quadratic Inequality Calculator is most useful when you treat it as a learning check, not just a shortcut to the final answer. Start by copying the original problem carefully, including signs, exponents, decimal points, fractions, parentheses, and units. Then enter the values in the same order the problem gives them. A small typing change can completely change the result, especially in algebra, statistics, geometry, and probability problems.

Before you press the button to calculate, make a quick estimate or prediction. The estimate does not need to be exact. Its job is to help you notice impossible answers. If a distance becomes negative, a probability is bigger than 1, an angle looks too large, or a decimal point seems misplaced, go back and check the input before trusting the final result.

Before you enter the problem

• Rewrite the problem in a clean line so every value is easy to see.
• Use parentheses around grouped expressions, especially in fractions and exponents.
• Keep units with the numbers while you work, even if the calculator only asks for the numbers.
• Check whether the problem wants an exact value, a decimal approximation, or a rounded answer.
• Look for restrictions such as positive values only, a chosen interval, or a required domain.

How to read the result

After the calculator gives a result, read more than the final line. If steps, tables, graphs, or intermediate values are shown, use them to understand how the answer was built. That is especially important when you are studying for a quiz or test, because teachers often give more credit for a correct process than for an unsupported number.

Try to identify the main idea behind the result. For example, ask yourself which formula was used, which operation changed the expression, which value controlled the graph, or which assumption made the answer possible. When you can explain that idea in your own words, the tool has helped you learn the skill instead of only checking one problem.

Common mistakes to avoid

Turn the answer into practice

One good way to study is to solve the problem by hand first, then use the Quadratic Inequality Calculator to check your work. If your answer is wrong, do not erase everything immediately. Find the first line where your work stops matching the calculator’s logic. That line is usually where the real misunderstanding happened.

1. Work the problem on paper and circle your final answer.
2. Use the calculator to check the result.
3. If the answers match, write one sentence explaining the method.
4. If they do not match, compare each step until you find the first difference.
5. Redo a similar problem without the tool to make sure the correction sticks.

When to use a calculator and when to work by hand

Use the Quadratic Inequality Calculator when you want to check a long calculation, explore a pattern, test a graph, or confirm a result after practicing. Work by hand when the assignment asks for steps, when you are learning a new method, or when the test will not allow a digital tool. The strongest students use both: hand work to build understanding and calculators to check accuracy.

If you are preparing for a timed test, practice some problems without the tool and some with it. That balance helps you build speed while still understanding the math. Over time, the goal is to need the calculator less often for routine steps and use it more strategically for checking, exploring, and confirming your reasoning.

Quick accuracy checklist

Before you leave the page, use the Quadratic Inequality Calculator result to answer three quick questions: What was the input? What rule, formula, or method connects the input to the result? What would change if one number in the problem changed? These questions turn a single calculator answer into a reusable math skill.

For homework, write the original problem, the calculator’s result, and one short explanation of the method in your notebook. For test review, cover the result and try to reproduce the setup on your own. If you can set up a similar problem without looking back at the calculator, you are much closer to owning the skill.