Calculator Inputs
Interactive Plot
The graph shows y = ax² + bx + c. Blue shaded x-ranges satisfy the chosen inequality.
Example Data Table
| Expression | Roots | Solution |
|---|---|---|
| x² - 5x + 6 > 0 | 2, 3 | x < 2 or x > 3 |
| x² - 4x + 4 ≤ 0 | 2 | x = 2 |
| x² + 2x - 3 ≥ 0 | -3, 1 | x ≤ -3 or x ≥ 1 |
| -x² + 9 < 0 | -3, 3 | x < -3 or x > 3 |
| 2x² + 8x + 10 > 0 | No real roots | All real numbers |
Formula Used
Start with the inequality ax² + bx + c relation 0. Compute the discriminant Δ = b² - 4ac and the roots x = (-b ± √Δ) / 2a.
If Δ is positive, the parabola crosses the x-axis twice. If Δ is zero, it only touches once. If Δ is negative, it never crosses the x-axis.
The sign of a tells how the parabola opens. When a is positive, values outside the real roots are positive. When a is negative, that sign pattern reverses.
How to Use This Calculator
- Enter coefficients a, b, and c for the expression ax² + bx + c.
- Choose the inequality symbol: >, ≥, <, or ≤.
- Select how many decimal places you want in the displayed roots and interval boundaries.
- Press Solve Inequality to place the result card above the form.
- Review the solution set, interval notation, roots, vertex, steps, and graph shading.
- Use the CSV or PDF buttons to export the current result summary.
FAQs
1. What is a quadratic inequality?
A quadratic inequality compares a quadratic expression to zero using >, ≥, <, or ≤. The goal is to find every x-value that makes the statement true.
2. Why are roots important here?
Roots mark where the expression changes sign or touches the axis. They split the number line into test intervals, which makes the final solution easy to describe.
3. What does the discriminant tell me?
The discriminant shows how many real roots exist. Positive means two real roots, zero means one repeated root, and negative means no real roots.
4. Why do some answers use unions of intervals?
A parabola can satisfy an inequality in two separate x-regions. That is why solutions often appear as two intervals joined by a union symbol.
5. When do I include the roots in the answer?
Include roots only when the inequality allows equality, such as ≥ or ≤. For strict inequalities, the roots are excluded because they make the expression exactly zero.
6. What happens when a equals zero?
The problem stops being quadratic and becomes linear or constant. This calculator still handles that case and returns the correct real-number solution set.
7. Why is the graph useful?
The graph lets you see where the parabola sits above or below the x-axis. The shaded bands immediately show which x-values satisfy the inequality.
8. Can I use decimals for coefficients?
Yes. The calculator accepts integers and decimals for all coefficients. You can also control the displayed precision for roots, intervals, and vertex values.