How to use
Use this calculator to solve a quadratic inequality in one real variable after moving all terms to the left: ax² + bx + c compared with zero. It answers which values of x work, rather than merely listing equation roots. Choose < or ≤ to find where the expression is negative or nonpositive, and > or ≥ for positive or nonnegative values. This is useful for checking algebra exercises and finding parameter ranges in simple quadratic models.
Replace the editable example values with your own data, choose the comparison or dimension where available, and press Calculate. Enter submits the same form. The result is confirmed only after this action. Editing a number or changing a selection removes the previous result and disables Copy result, so an old answer cannot be mistaken for the current calculation. Clear empties the number fields while keeping the selected mode. After a successful calculation, Copy result copies the input and the numerical output as plain text. If clipboard access is blocked, select the visible text instead. Pale values are a runnable example. Focusing a number field clears all example numbers once. Your later entries are kept. Input preview — calculate to confirm the result.
How it works and reading the result
When a is nonzero, the discriminant b² − 4ac determines the real boundaries. Two distinct roots divide the line into three sign intervals. The outer intervals have the sign of a, and the middle has the opposite sign. A repeated root touches zero without changing sign. With no real roots, the sign stays the same everywhere. The discriminant sign is calculated from the entered decimal coefficients using integer arithmetic, avoiding an arbitrary small-number cutoff.
Read the solution set first, then compare it with the sign intervals. The intervals in the sign summary are open because roots are checked separately. A strict inequality excludes every zero; ≤ and ≥ may include them. Square brackets and filled circles mean inclusion, while parentheses and hollow circles mean exclusion. The curve plots y = ax² + bx + c with labelled axis ranges. It is a finite viewing window, while the number-line arrows describe intervals extending beyond that window. Axes can have different scales.
Examples
Example 1: a = 1, b = −3, c = 2 and ≤ gives [1; 2]. The roots are 1 and 2 and the sign pattern is +, −, +. At x = 1.5 the expression is −0.25, so it belongs to the solution. The boundary substitutions give zero. Choosing ≥ instead yields (−∞; 1] ∪ [2; ∞).
Example 2: a = 1, b = −4, c = 4 and < gives ∅ because the expression is (x − 2)². Switching to ≤ gives the single point {2}. Switching to > gives every real number except 2. A double root is therefore not evidence that the sign changes across the boundary.
Limits and troubleshooting
The tool accepts decimal coefficients, not symbolic expressions, fractions or inequalities involving y as a second unknown. Inputs are zero or magnitudes from 1e−12 to 1e12. If a = 0, the tool solves the remaining linear inequality; if b is also zero, it tests the constant. Displayed roots and substitutions are numerical approximations with extra digits to preserve endpoint meaning. Very close roots or extreme scale differences may be hard to distinguish visually. Read the numeric solution and original coefficients rather than estimating a root from the drawing.
Use at most 15 significant input digits. Boundaries too close to distinguish numerically are rejected; re-express the variable if necessary.
Frequently asked questions
Why are there no roots but still a solution?
For x² + 1 > 0, every x works although the curve never reaches zero. Roots and solution sets answer different questions.
What happens with a negative leading coefficient?
The curve opens downward and its outer sign intervals are negative. The selected comparison still decides which intervals belong.
How is a linear case handled?
For a = 0, b = −2, c = 6 and >, the condition becomes −2x + 6 > 0, so the answer is (−∞; 3).
What does a tiny substitution residual mean?
A root was approximated numerically. A small residual checks consistency but is not a proof that the displayed decimal is an exact algebraic root.