Quadratic Inequality Solver

Introduction: solving quadratic inequalities step by step

Solving a quadratic inequality is less about producing a single number and more about locating every real x-value that makes ax² + bx + c sit above, below, or on zero. That is exactly what this quadratic inequality solver does: you enter the coefficients, choose the relation, and the calculator returns the real intervals that satisfy the inequality.

This page is useful when you want a clean algebra checkpoint before you hand in homework or compare two versions of the same parabola. The explanation below shows how the sign of a changes the opening of the parabola, how the discriminant tells you whether the quadratic crosses the axis, and how interval notation records the final answer.

Keep one central idea in mind: a quadratic inequality is not solved by guessing one x-value. Instead, its roots split the number line into regions, and each region has a consistent sign until another root is crossed.

What quadratic inequality does this calculator solve?

This quadratic inequality solver finds which real x-values make ax² + bx + c greater than, less than, greater than or equal to, or less than or equal to zero. It translates that algebraic question into a solution set written in interval notation, so you can see whether the answer is one interval, two outward intervals, all real numbers, or no solution.

Before calculating, put the inequality in standard form with zero on one side. If you begin with a graph, a factored expression, or a word problem, convert it carefully into ax² + bx + c first. This keeps the sign of every coefficient tied to the expression you actually intend to solve.

How to use this quadratic inequality solver

This quadratic inequality solver works best when you enter one complete polynomial and one comparison symbol, then read the result as one algebra answer rather than three unrelated numbers.

1. Enter Coefficient a, the coefficient of x² in ax² + bx + c.
2. Enter Coefficient b, the coefficient of x.
3. Enter Coefficient c, the constant term.
4. Choose the comparison sign against zero. The symbol determines whether boundary roots count in the final answer.
5. Select Solve, then check whether the result is an open interval, a closed interval, a union, all real numbers, or no solution.

If you are comparing practice problems, keep a note of the coefficients and relation you used. A changed minus sign can produce a completely different parabola and solution set.

Quadratic coefficients: how each value changes the graph

The form collects the three coefficients and comparison sign that define the quadratic inequality. Most mistakes come from moving terms to the wrong side of the inequality or losing a negative sign. Coefficient a controls whether the parabola opens upward or downward; b shifts the axis of symmetry and changes the root locations; c is the y-intercept and moves the graph vertically.

Check the expression as written before entering it. For example, −2x² + 3x − 5 has a = −2, b = 3, and c = −5. If a is zero, the expression is not quadratic, and this page intentionally switches to its linear-inequality logic instead.

Quadratic inequality formulas: discriminant, roots, and signs

Quadratic inequality formulas begin with the discriminant. After moving all terms to one side, calculate D to determine how many real roots can divide the number line:

When D is positive, there are two distinct real roots. When D is zero, the parabola touches the x-axis at one repeated root. When D is negative, there are no real roots, so a nonzero quadratic stays entirely above or entirely below the x-axis. For two roots, the calculator uses:

From there, the sign of a supplies the sign-chart pattern. An upward-opening parabola is positive outside two distinct roots and negative between them. A downward-opening parabola reverses that pattern. Strict symbols, > and <, use open endpoints; inclusive symbols, ≥ and ≤, include roots where the expression equals zero.

Worked quadratic inequality example: x² − 5x + 6 ≥ 0

Consider x² − 5x + 6 ≥ 0. Enter a = 1, b = −5, c = 6, and choose ≥ 0. The expression factors as (x − 2)(x − 3), giving roots at x = 2 and x = 3.

Because a is positive, the parabola opens upward. Its values are nonnegative on the outside of the roots and negative between them. The inclusive symbol keeps both roots, so the solution set is (−∞, 2] ∪ [3, ∞).

Changing only the relation demonstrates the sign chart: > 0 gives (−∞, 2) ∪ (3, ∞), while < 0 gives (2, 3). The polynomial does not change; only the region that counts as a solution changes.

Comparison table: how the inequality sign changes the solution set

This compact table keeps x² − 5x + 6 fixed and changes only the comparison. It shows why equality changes the brackets but not the positive-versus-negative regions.

For another polynomial, use the calculator’s discriminant and roots rather than copying this particular table. The direction of the parabola and your selected symbol always determine which side of the roots is included.

How to interpret the quadratic solution set

The results panel gives the discriminant, any real roots, and final interval notation. Read it as an algebra summary. First confirm that the selected inequality sign is correct. Next, check whether the roots are plausible for the coefficients. Finally, ask whether an upward or downward opening parabola should make the chosen interval positive or negative.

Open parentheses exclude a boundary because a strict inequality cannot include a point where the expression equals zero. Square brackets include a root. The displayed roots are rounded to four decimals, so an exact radical answer from hand work may look slightly different on screen even though it represents the same location.

Limitations and assumptions for quadratic inequality solving

Quadratic inequality solving on this page assumes real-number inputs and reports real-number intervals. The tool reads a, b, and c literally, so swapping coefficients or changing a sign changes the polynomial. If a = 0, it solves the resulting linear inequality; if both a and b are zero, it determines whether the constant statement is always true or never true.

Roots and endpoints are rounded for display, while the calculation uses the entered numeric values. Teacher-specific formatting, exact radical notation, graph sketches, and restrictions such as integer-only domains are not imposed automatically. For coursework or another high-stakes use, verify the setup with hand algebra and test a sample x-value from each reported interval.