What does it mean when there are no real roots?

It means the parabola never touches the x-axis, which happens when the discriminant b² − 4ac is negative. The equation still has two solutions, but they involve the square root of a negative number and are written as a complex conjugate pair such as 2 ± 3i. If your course has not covered complex numbers yet, "no real roots" is the expected answer and the graph makes it obvious why — the whole curve sits above or below the axis.

Why does the answer show a long decimal instead of a square root?

This is a numerical solver, not a symbolic one. It computes values rather than manipulating expressions, so a root of √5 comes back as 2.2360679775 rather than staying in radical form. For checking a result that is usually what you want. If your assignment requires the exact surd, use the steps shown to reconstruct it — the discriminant line gives you the number under the root sign directly.

Can it solve equations with three unknowns, or quartics?

Not currently. Simultaneous equations are limited to two equations in two unknowns, and single equations go up to degree three. Quartics do have a closed-form solution but it is long enough that a numerical root-finder is the more practical tool, and systems with three or more unknowns are better served by a matrix calculator. If you need one of those, say so — demand is how the tool list gets decided here.

Why does the cubic sometimes give three roots and sometimes one?

Every cubic has three roots counted with multiplicity, but they are not always all real. A cubic curve always crosses the x-axis at least once, which is why there is never a cubic with no real roots at all, but it may cross once and turn back without reaching the axis again — in that case one root is real and the other two are a complex pair, and only the real one is listed. When the curve crosses three times, all three are real and all three are shown.

What if the determinant of my simultaneous equations is zero?

Then the two lines are parallel and Cramer's rule cannot give a unique answer. There are two sub-cases and the tool distinguishes them: if the equations describe the same line, every point on it satisfies both and there are infinitely many solutions; if they describe genuinely parallel but different lines, they never intersect and there is no solution. Getting "no solution" is a valid result, not an error.

Is the leading coefficient allowed to be zero?

No, and the tool blocks it rather than dividing by zero quietly. If a is zero in ax² + bx + c = 0 there is no x² term, so it is not a quadratic at all — it is a linear equation and the linear tab will solve it correctly. The same applies to the cubic tab. This is the single most common cause of a solver returning infinity or NaN, which is why it is caught up front.

Does this tool upload my data?

No. Everything runs right in your browser, so your data never leaves your device — it even works offline once the page has loaded.

🧮 Equation Solver — With the Working Shown

Enter the coefficients, press Solve, and you get the roots together with the steps that produced them. For a quadratic you also get the discriminant, the vertex, the axis of symmetry, the factored form, and a graph with the roots marked where the curve crosses the x-axis. Most solvers hand back a number; the working is usually the part you actually needed, especially if you are checking homework rather than just getting through it.

An equation solver finds the values of the unknown that make an equation true. This one handles linear (ax + b = 0), quadratic (ax² + bx + c = 0), cubic (ax³ + bx² + cx + d = 0) and two-variable simultaneous equations, and shows the algebra it used rather than only the answer.

ax² + bx + c = 0

Private by design. This tool runs entirely in your browser — nothing you enter is uploaded or stored, and it works offline.

About

The discriminant is the part worth understanding, because it tells you what kind of answer to expect before you finish solving. For ax² + bx + c = 0 the value b² − 4ac decides everything: positive means the parabola crosses the x-axis twice and there are two distinct real roots, zero means it touches the axis at exactly one point and the two roots coincide into a repeated root, and negative means the curve never reaches the axis and both roots are complex, appearing as a conjugate pair. That single number is why a graph and an algebraic answer always agree — they are describing the same fact from two directions.

The cubic case contains a trap that catches most naive implementations. Cardano's formula, the standard closed-form solution, requires taking the square root of a quantity that goes negative precisely when the cubic has three real roots. This is the casus irreducibilis: the honest algebraic path to three perfectly real answers runs through imaginary numbers. A solver that applies Cardano directly will produce NaN or silently wrong output on exactly the inputs a student is most likely to type. This tool detects that branch and switches to the trigonometric solution, using arccosine to extract all three real roots without ever leaving the reals.

Simultaneous equations are solved with Cramer's rule, and the determinant carries the same diagnostic role the discriminant does for quadratics. If the determinant is non-zero the two lines cross at exactly one point and there is a unique solution. If it is zero the lines are parallel, and the follow-up question is whether they are the same line: if so every point satisfies both equations and there are infinitely many solutions, and if not the lines never meet and there is no solution at all. Reporting "no solution" and "infinitely many solutions" as different outcomes matters, because they look identical to a solver that only checks whether the determinant vanished.

One implementation detail affects what you see. Binary floating point cannot represent most decimal fractions exactly, so an arithmetically clean root can arrive as 0.30000000000000004 or 2.9999999999999996. Results are rounded at the tenth decimal place before display, which removes that noise without hiding genuine precision — a root that really is 2.9999 will still show as 2.9999. Graph ranges are chosen from the roots rather than fixed, so the interesting part of the curve is always on screen instead of the roots sitting somewhere off the edge.

The limits are worth stating. This solves numerically with real coefficients, so it will not do symbolic algebra, keep answers as exact surds or fractions, handle equations with letters as parameters, or go beyond degree three. For a quadratic whose roots are irrational you get the decimal value, not an expression with a radical sign in it.

How to use

Choose the equation type — quadratic, linear, cubic or simultaneous. The form at the top of the panel shows exactly which coefficient goes in which box.
Enter the coefficients including their signs. For 2x² − 5x + 3 = 0 that is a = 2, b = −5, c = 3. A missing term is a zero, so x² − 4 = 0 means b = 0.
Press Solve. The roots appear first, then the numbered steps, then the extra values — discriminant, vertex, factored form and the sum and product of the roots.
Check the graph. Where the curve meets the x-axis is where the roots are, so if the marked points do not sit on the crossings you have mistyped a coefficient.

FAQ

What does it mean when there are no real roots?: It means the parabola never touches the x-axis, which happens when the discriminant b² − 4ac is negative. The equation still has two solutions, but they involve the square root of a negative number and are written as a complex conjugate pair such as 2 ± 3i. If your course has not covered complex numbers yet, "no real roots" is the expected answer and the graph makes it obvious why — the whole curve sits above or below the axis.
Why does the answer show a long decimal instead of a square root?: This is a numerical solver, not a symbolic one. It computes values rather than manipulating expressions, so a root of √5 comes back as 2.2360679775 rather than staying in radical form. For checking a result that is usually what you want. If your assignment requires the exact surd, use the steps shown to reconstruct it — the discriminant line gives you the number under the root sign directly.
Can it solve equations with three unknowns, or quartics?: Not currently. Simultaneous equations are limited to two equations in two unknowns, and single equations go up to degree three. Quartics do have a closed-form solution but it is long enough that a numerical root-finder is the more practical tool, and systems with three or more unknowns are better served by a matrix calculator. If you need one of those, say so — demand is how the tool list gets decided here.
Why does the cubic sometimes give three roots and sometimes one?: Every cubic has three roots counted with multiplicity, but they are not always all real. A cubic curve always crosses the x-axis at least once, which is why there is never a cubic with no real roots at all, but it may cross once and turn back without reaching the axis again — in that case one root is real and the other two are a complex pair, and only the real one is listed. When the curve crosses three times, all three are real and all three are shown.
What if the determinant of my simultaneous equations is zero?: Then the two lines are parallel and Cramer's rule cannot give a unique answer. There are two sub-cases and the tool distinguishes them: if the equations describe the same line, every point on it satisfies both and there are infinitely many solutions; if they describe genuinely parallel but different lines, they never intersect and there is no solution. Getting "no solution" is a valid result, not an error.
Is the leading coefficient allowed to be zero?: No, and the tool blocks it rather than dividing by zero quietly. If a is zero in ax² + bx + c = 0 there is no x² term, so it is not a quadratic at all — it is a linear equation and the linear tab will solve it correctly. The same applies to the cubic tab. This is the single most common cause of a solver returning infinity or NaN, which is why it is caught up front.
Does this tool upload my data?: No. Everything runs right in your browser, so your data never leaves your device — it even works offline once the page has loaded.