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 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 most helpful when you want a clean algebra checkpoint before you hand in homework or compare two versions of the same parabola. The notes below explain 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 encodes the final answer.
As you read on, keep one idea in mind: a quadratic inequality is not solved by guessing a value, but by splitting the number line around the roots and checking which side meets the comparison you chose.
What quadratic inequality does this calculator solve?
For this quadratic inequality solver, the question is always which real x-values make ax² + bx + c greater than, less than, greater than or equal to, or less than or equal to zero. The calculator translates that algebraic question into a solution set you can read directly in interval notation.
Before you calculate, write the inequality in standard form so the coefficients are easy to identify. If you start from a graph, a factored expression, or a word problem, convert it carefully into ax² + bx + c first; that way the sign of each coefficient matches the expression you actually mean to solve.
When the inequality is already in standard form, the main choices are straightforward: pick the correct comparison sign, verify whether the leading coefficient is positive or negative, and let the roots divide the real line into regions that can be tested.
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 a single algebra answer.
- Enter Coefficient a as the quadratic term in ax² + bx + c.
- Enter Coefficient b as the linear coefficient paired with x.
- Enter Coefficient c as the constant term that shifts the parabola vertically.
- Choose the comparison sign against zero; the inequality symbol decides whether the boundary roots count as part of the answer.
- Run the calculation to refresh the solution intervals.
- Check whether the answer is an open interval, a closed interval, a union, all real numbers, or no solution before you copy it elsewhere.
If you are comparing homework attempts or practice problems, save the coefficients and the relation you used so you can reproduce the same quadratic inequality later without reworking the setup.
Quadratic coefficients: how to pick good values
The calculator’s form collects the three coefficients and the comparison sign that define the quadratic inequality. Most mistakes come from sign errors, moving terms to the wrong side of the inequality, or solving a different polynomial than the one written in the problem. Use the following checklist as you enter the values:
- Signs: verify whether the leading coefficient opens the parabola upward or downward, and make sure the linear and constant terms keep their original signs.
- Ranges: if your class or model restricts the values of a, b, or c, keep your inputs inside that intended range.
- Defaults: any prefilled values are only examples; replace them with the coefficients from your own quadratic inequality before trusting the answer.
- Consistency: make sure the expression you enter is the same one you would solve by hand after factoring, expanding, or rearranging.
Common inputs for a quadratic inequality problem include:
- Coefficient a: the leading term that decides whether the parabola opens upward or downward and whether the solution lies inside or outside the roots.
- Coefficient b: the x-term that shifts the axis of symmetry and moves the roots left or right.
- Coefficient c: the constant term that raises or lowers the parabola and sets the y-intercept.
- Compare ax² + bx + c to zero: the relation symbol that tells the solver whether the boundary points belong in the final interval.
If you are unsure about a coefficient, work backwards from the roots, from a graph, or from a factored form and translate it carefully into standard form. Running a second scenario with a different sign or coefficient can help you catch transcription mistakes before you hand in the answer.
Quadratic formulas: how the solver turns coefficients into intervals
Quadratic inequality solvers follow the same core algebra every time: move everything to one side, inspect the discriminant D = b² - 4ac, and use the real roots to divide the number line into test intervals. That is the logic this calculator uses when it turns your coefficients into a final solution set.
If D is positive, the quadratic has two distinct real roots; if D is zero, it touches the x-axis once; and if D is negative, the parabola never crosses the axis. From there, the sign of a tells you whether the expression is positive outside the roots or between them, and the comparison sign tells you whether the roots themselves belong to the answer.
That pattern is why the same quadratic can produce several different answers depending on whether you ask for > 0, ≥ 0, < 0, or ≤ 0. The calculator is not changing the polynomial; it is changing which regions of the real line count as a solution.
Worked quadratic inequality example: x² - 5x + 6 ≥ 0
A worked quadratic inequality example is the fastest way to see how the coefficients become intervals. Suppose you enter a = 1, b = -5, c = 6, and choose ≥ 0. The quadratic factors as (x - 2)(x - 3), so the real roots are x = 2 and x = 3.
Because the leading coefficient is positive, the parabola opens upward. That means the expression is nonnegative on the outside of the roots and negative between them. Since the comparison sign includes equality, the boundary points stay in the answer.
Solution set: (-∞, 2] ∪ [3, ∞)
Notice how this example mirrors the result panel: first you identify the roots, then you decide whether the intervals are open or closed, and finally you combine the pieces into the full real-number solution. If you change the comparison sign to > 0, the same example becomes (-∞, 2) ∪ (3, ∞); if you change it to < 0, the answer becomes (2, 3). That is the central pattern this calculator is built to show.
Comparison table: how the inequality sign changes the solution set
The table below keeps the same quadratic, x² - 5x + 6, and changes only the comparison sign so you can see exactly how the solution set moves from outside the roots to inside them. It is a better guide than a generic sensitivity table because every row corresponds to a real quadratic inequality.
| Comparison sign | Leading coefficient a | b and c | Solution set | How to read it |
|---|---|---|---|---|
| > 0 | 1 | b = -5, c = 6 | (-∞, 2) ∪ (3, ∞) | Strict positivity lives outside the roots. |
| ≥ 0 | 1 | b = -5, c = 6 | (-∞, 2] ∪ [3, ∞) | Zero is included, so the roots stay in the answer. |
| < 0 | 1 | b = -5, c = 6 | (2, 3) | The expression is negative only between the roots. |
| ≤ 0 | 1 | b = -5, c = 6 | [2, 3] | Equality makes the boundary values part of the interval. |
Use the calculator's actual result panel to check any other polynomial: the sign of a and the comparison symbol together determine whether the answer opens outward, sits between the roots, or collapses to a repeated boundary point.
How to interpret the quadratic solution set
The results panel gives you the discriminant, any real roots, and the final interval notation, so read it as an algebra summary rather than a raw calculation log. When you get an answer, ask three questions: (1) does the inequality sign match the one you chose? (2) are the roots plausible for these coefficients? (3) does the sign of the parabola produce the interval you expected? If you can answer yes to those, the solution set is ready to use.
Open parentheses mean the boundary point is excluded because the inequality is strict, while square brackets mean the root itself belongs to the answer. The displayed roots are rounded to four decimals, so tiny differences from hand work are normal.
If you want to keep a record, use the Copy Result button to copy the interval text or save a screenshot for your notes. The value of the calculator is that it shows the full quadratic inequality process in one place: coefficients in, discriminant and roots out, interval notation at the end.
Limitations and assumptions for quadratic inequality solving
No quadratic inequality solver can replace the algebra behind the answer. This tool is fast and transparent, but it still depends on a few assumptions and display choices. Keep these in mind:
- Input interpretation: read a, b, and c literally; swapping them changes the polynomial and therefore the solution interval.
- Linear fallback: if a = 0, the page treats the problem as linear, so make sure that matches the expression you intended to solve.
- Rounding: roots and interval endpoints are rounded to four decimals, so a hand-solved exact radical may look slightly different on screen.
- Real-number scope: the output describes real intervals only; if your class expects a different notation or a graph-based explanation, convert the result accordingly.
- Special conventions: teacher-specific formatting rules or symbolic forms are not enforced automatically, so verify the presentation before you submit it.
If you use the answer for coursework, grading, or another high-stakes setting, verify it with hand algebra or a second method. The value of this calculator is that it makes the quadratic inequality process visible: coefficients in, discriminant and roots out, interval notation at the end.