Build the inequality in the display at the top: click the Var, Num and Op buttons, the ^ or x² power buttons, one of the four Ineq symbols and the brackets, or type on the keyboard. Learn more about building inequalities
Clear empties the display, the curved arrow undoes the last edit and the ⌫ button deletes the character before the caret. Learn more about editing
Click Solve, or press Enter in the display, once the inequality is complete; an expression without an inequality symbol or without a squared term is refused. Learn more about solving
The right panel lists the steps from standard form through the discriminant and the roots to the sign chart, and the answer card gives the inequality, a note and the interval; the box under the solver explains the method. Learn more about the solution
A negative discriminant ends in All real numbers or No solution; a zero discriminant in a single point or all reals except one point. Learn more about these cases
x²
Solve quadratic inequalities step by step
Enter inequality...
Var
Num
Op
Ineq
Select an inequality type or enter your own, then click Solve to see the step-by-step solution.
How the solver works
The solver reads one quadratic inequality in x, y or n with <, >, leq or geq, moves every term to the left, multiplies by −1 and flips the sign if the leading coefficient is negative, and computes the discriminant. Two roots give a sign chart with the solution between or outside them; one repeated root gives a single point or everything but that point; no real root gives all real numbers or no solution. The answer is shown as an inequality and as an interval. Learn more about quadratic inequalities
The solver shows an inequality display at the top with a blinking yellow caret marking the cursor position. Click anywhere in the display to place the cursor, or use arrow keys, Home, and End for keyboard navigation. Type directly or use the button panel below.
Three quick experiments:
• Type x2−5x+6<0 and press Enter — the right panel finds roots x=2 and x=3, sketches the upward parabola, and reports the between-the-roots solution 2<x<3 in the interval (2,3). • Click the "No Real Roots" example template — an inequality like x2+5>0 loads. The discriminant is negative, the parabola sits above the x-axis, and the answer is all real numbers. • Try the "Negative Leading" template — an inequality with −x2. The solver multiplies both sides by −1, flips the direction, then proceeds as a standard upward-parabola problem.
The Solve button is disabled until you enter something. The dedicated "Ineq" row inserts the four inequality symbols; the x2 shortcut button inserts the power after a variable you have already typed. An expression without an inequality symbol is refused with "Input must be an inequality (use <, >, ≤, or ≥)", and one without a squared term with a message that points to the linear inequality solver. What makes an inequality quadratic is explained in what is a quadratic inequality.
Building Inequalities with Buttons and Keyboard
DemoA negative leading coefficient, undo, delete, Solve
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The button panel groups inputs by type. All inputs can be made either by clicking or typing on the keyboard.
• Variable row — x, y, n. • Number row — digits 0 through 9 and the decimal point. • Operator row — multiplication, division, plus, minus. • Ineq row — the four inequality symbols, highlighted in amber. • Special row — the caret operator for arbitrary exponents, the x2 shortcut button, and parentheses for grouping.
The x2 button inserts a caret followed by a 2 (which the display renders as a superscript). For higher powers you can type the caret followed by any number, but the solver only handles quadratic expressions; cubic or higher will produce an error.
Keyboard shortcuts: type letters and digits directly, use star or slash for multiplication and division, type < and > directly, type <= or >= to insert the combined symbols ≤ and ≥, and press Enter to solve. Ctrl+Z undoes up to fifty edits back. The Clear button empties the display entirely; the curved-arrow button steps through the undo stack one edit at a time; the ⌫ button deletes the character before the caret. A negative leading coefficient, as in −x2+4>0, is handled by the Multiply by −1 step, which flips the sign and gives (−2,2). Why the sign of the leading coefficient matters is explained in the parabola method. The Unicode characters x2 and x3 are recognized directly if pasted.
Try an Example — Eight Form Templates
DemoOpen the templates, load two forms
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Click the "Try an Example" header to expand the template panel. Each card generates a random inequality of that form. Clicking again produces a new random version.
• Standard (less than 0) — x2+bx+c<0. Two real roots with the solution sitting between them. • Standard (greater than 0) — x2+bx+c>0. Two real roots with the solution sitting outside as a union of two rays. • Non-strict (≤0) — like the strict less-than version but with closed endpoints (brackets in the interval). • Non-strict (≥0) — like the strict greater-than version but with closed endpoints in the union. • Leading coefficient not 1 — ax2+bx+c<0 with a=1. Tests the divide-by-a structural path. • Negative leading — −x2+bx+c>0. Exercises the multiply-by-−1 step that flips the inequality direction. • No real roots — x2+c>0 with positive c. The discriminant is negative and the answer collapses to all reals or no solution depending on the direction. • Non-zero RHS — x2+bx<c with c=0. Tests the move-everything-to-the-left step before standard analysis.
Roughly 80 percent of generated inequalities produce clean integer roots. Loading a template clears the previous solution; click Solve to work the new inequality. The five stages applied to every template are listed in the solving process explained.
Reading the Step-by-Step Solution
DemoA single point, then a union outside the roots
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The solution panel shows each algebraic move as a labeled step. The sequence depends on the discriminant.
• Multiply by −1 (when applicable) — if the leading coefficient is negative, the solver multiplies both sides by −1 and flips the inequality direction. From this point the analysis uses a positive leading coefficient. • Standard Form — restates the inequality with everything on the left and zero on the right. • Calculate Discriminant — shows the value of Δ=b2−4ac. • No Real Roots (when Δ<0) — reports that the parabola never crosses the x-axis and gives the answer (all reals or no solution). • Repeated Root (when Δ=0) — reports the single root and analyzes whether the inequality holds at that point only, everywhere except that point, everywhere, or nowhere. • Find Roots (when Δ>0) — reports the two distinct roots from the quadratic formula. • Sign Chart — states which intervals are positive and which are negative based on the upward-opening parabola. • Solution Interval — selects the matching interval(s) for the inequality direction.
The final-answer card shows the answer in three forms: the inequality form (such as r1<x<r2), a plain-language note (between or outside the roots), and the interval notation. x2+4x+4≤0 has Δ=0 and holds only at the single point x=−2, written {−2}; x2−x−6≥0 has the roots −2 and 3 and the outside solution (−∞,−2]∪[3,∞). The box under the solver, How the solver works, summarises the method in a few lines and links to the theory below. The sign pattern behind every answer is explained in the parabola method.
The Parabola Method — Why It Works
Every quadratic inequality reduces to a sign question about a parabola.
Setup. Move all terms to one side so the inequality becomes ax2+bx+c compared to 0. The left side is a quadratic function whose graph is a parabola. The inequality asks: for which x is the parabola above zero, below zero, equal to zero, or non-negative or non-positive?
Two facts determine everything.
• Direction of opening. If a>0 the parabola opens upward; if a<0 it opens downward. The solver always normalizes to a>0 by multiplying by −1 and flipping direction if needed. • Roots. The roots (x-intercepts) are where the parabola crosses zero. The discriminant determines how many: two distinct, one repeated, or none.
Sign pattern for an upward-opening parabola.
• Two distinct roots r1<r2: negative on (r1,r2), positive on (−∞,r1)∪(r2,∞), zero at r1 and r2. • One repeated root r: zero at r, positive everywhere else. Never negative. • No real roots: always positive. Never zero or negative.
Reading the answer. Match the inequality direction to the matching sign region. Strict inequalities exclude the roots; non-strict include them.
• f(x)<0 with two roots: open interval between roots, (r1,r2). • f(x)≤0 with two roots: closed interval, [r1,r2]. • f(x)>0 with two roots: open union, (−∞,r1)∪(r2,∞). • f(x)≥0 with two roots: closed union, (−∞,r1]∪[r2,∞).
For deeper coverage see quadratic function and parabola.
A quadratic inequality is a comparison between a quadratic expression and zero (or any other value, after rearrangement) using one of the four symbols <, >, ≤, ≥. The standard form is
ax2+bx+c□0
where □ is one of the four symbols and a, b, c are real numbers with a=0. The variable x appears squared but to no higher power.
Solution shapes. Unlike linear inequalities whose solutions are always rays, quadratic inequalities produce six possible shapes depending on the discriminant and direction:
• A single interval between the roots — f(x)<0 or ≤0 with a>0 and two real roots. • A union of two rays outside the roots — f(x)>0 or ≥0 with a>0 and two real roots. • A single point — non-strict inequality at a repeated root. • All reals except one point — strict inequality at a repeated root in the always-positive direction. • All real numbers — the parabola sits entirely on the correct side of zero. • No solution — the parabola sits entirely on the wrong side of zero.
Discriminant decides the case. The number Δ=b2−4ac classifies which solution shape applies. Positive discriminant means two distinct roots and gives between or union solutions. Zero discriminant means one repeated root and gives point or all-except-point solutions. Negative discriminant means no real roots and gives all-reals or no-solution.
For deeper coverage see quadratic inequality, discriminant, and interval notation.
Five stages handle any quadratic inequality. The solver follows this exact sequence.
• Stage 1: Reduce to standard form. Move all terms to the left so the right side is zero. Collect x2, x, and constant coefficients into a, b, c.
• Stage 2: Normalize sign of a. If a<0, multiply both sides by −1 and flip the inequality direction. From here the parabola opens upward and the sign pattern is the standard one.
• Stage 3: Compute the discriminant.Δ=b2−4ac. Branch on the sign of Δ.
• Stage 4: Find the roots if any.
• Δ>0: two distinct real roots r1<r2 from the quadratic formula. • Δ=0: one repeated root r=−b/(2a). • Δ<0: no real roots.
• Stage 5: Match sign pattern to inequality direction. The solver picks the interval(s) on the number line where the quadratic has the required sign, applying open or closed endpoints based on whether the inequality is strict.
The implementation. Internally, after collecting coefficients, the solver hands the problem to a sign-analysis routine that constructs the answer as one of six branch outputs: all reals, no solution, single point, all reals except a point, compound between-the-roots, or union outside-the-roots. The final-answer card adapts its rendering to the matched branch.
For comprehensive treatment see solving quadratic inequalities and sign analysis.
Quadratic Equation — the equality counterpart ax2+bx+c=0. The roots of the equation are the boundary points of the inequality solution set; the quadratic formula gives both.
Discriminant — the quantity Δ=b2−4ac. Positive means two real roots, zero means one repeated root, negative means no real roots. Drives every classification in quadratic inequality solving.
Parabola — the graph of a quadratic function y=ax2+bx+c. The inequality ax2+bx+c□0 asks where the parabola is above, below, or touching the x-axis.
Sign Chart — the tabular method of tracking signs across critical points. For quadratics there are at most two critical points (the roots), so the chart has at most three intervals.
Interval Notation — the standard way to express the solution. Parentheses for strict, brackets for non-strict, infinity always with parentheses, union symbol ∪ joining disconnected pieces.
Linear Inequality — the lower-degree counterpart. The solution is always a single ray. Linear inequalities are a degenerate case of quadratic where a=0.
Polynomial Inequality — the generalization to higher degrees. The sign-chart method extends directly, with one critical point per real root.
Vieta's Formulas — the relations between the roots and coefficients. For ax2+bx+c with roots r1 and r2: r1+r2=−b/a and r1r2=c/a. Useful for verifying root computations.
Completing the Square — an alternative way to find the vertex and roots of the parabola, sometimes more illuminating than the quadratic formula for inequality analysis.
Vertex — the turning point of the parabola at x=−b/(2a). For a non-strict inequality with discriminant zero, the vertex is exactly the root and the inequality may hold at this single point.