Enter the three coefficients a, b and c and this calculator solves ax² + bx + c = 0 instantly, giving real roots where they exist and complex conjugate roots where they do not — rather than stopping at "no solution". Alongside the answers it returns the discriminant and what it tells you, the vertex and axis of symmetry, the sum and product of the roots, the factored form, and the substituted quadratic-formula working so you can check each line. Pair it with the square root calculator for the radical in the formula, or the slope calculator for linear equations.
Quadratic Equation Calculator
Every root — real or complex — plus the vertex, factored form and the full quadratic-formula working.
For an equation in the form ax² + bx + c = 0
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Roots
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Discriminant (b² − 4ac)—
Nature of roots—
Vertex—
Axis of symmetry—
Sum of roots—
Product of roots—
Factored form
Working
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The quadratic formula
Any equation of the form ax² + bx + c = 0, where a is not zero, is solved by the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. The ± is why a quadratic normally has two answers — one taking the plus, one taking the minus. If a is zero the equation is not quadratic at all but linear, and it has a single root at x = −c/b.
The discriminant tells you what to expect
The part under the square root, b² − 4ac, is called the discriminant, and its sign decides everything before you finish the calculation. If it is positive, you get two distinct real roots and the parabola crosses the x-axis twice. If it is zero, both roots collapse into one repeated root and the parabola just touches the axis at its vertex. If it is negative, there are no real roots — the parabola misses the axis entirely — and the two solutions are a complex conjugate pair of the form p ± qi.
Complex roots are still answers
A negative discriminant does not mean "no solution", which is where many calculators stop unhelpfully. It means the solutions are complex numbers. √(−16) is 4i, so the roots come out as something like 2 + 4i and 2 − 4i. They always arrive as a conjugate pair with identical real parts and opposite imaginary parts, and this calculator gives them to you in full.
Vertex and axis of symmetry
Every parabola is symmetrical about a vertical line through its turning point. That line is the axis of symmetry, at x = −b ÷ 2a, and the turning point itself is the vertex. Substituting the axis value back into the equation gives the vertex y-coordinate. When a is positive the parabola opens upward and the vertex is the minimum; when a is negative it opens downward and the vertex is the maximum — which is exactly how you solve optimisation problems like maximum area or projectile peak height.
Sum and product of roots (Vieta's formulas)
You can read two facts straight off the coefficients without solving: the roots add to −b/a and multiply to c/a. This is a fast sanity check on any answer you have worked out by hand, and it also lets you build a quadratic from known roots — if the roots are 2 and 3, the equation is x² − 5x + 6 = 0.
Factored form
When the roots are r₁ and r₂, the quadratic factors as a(x − r₁)(x − r₂). Where the roots are whole numbers or simple fractions this gives clean factors like (x − 2)(x − 3). Irrational or complex roots still factor, just not neatly — which is exactly when the formula beats trying to factor by inspection.
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Frequently asked questions
What does a negative discriminant mean?
It means the parabola never crosses the x-axis, so there are no real roots. The equation still has two solutions, but they are complex conjugates of the form p + qi and p − qi. This calculator shows them rather than reporting no answer.
What if a equals zero?
Then it is not a quadratic — the x² term vanishes and you have a linear equation bx + c = 0 with a single root at x = −c/b. The calculator detects this and solves it as a linear equation instead.
How do I find the vertex of a parabola?
The x-coordinate is −b ÷ 2a. Substitute that back into the equation to get the y-coordinate. If a is positive the vertex is the lowest point; if a is negative it is the highest point.
When can I factor instead of using the formula?
Factoring is quickest when the discriminant is a perfect square, because that makes the roots rational. If the discriminant is not a perfect square, the roots are irrational and factoring by inspection will not work — use the formula.
What is a repeated root?
When the discriminant is exactly zero, both branches of the ± give the same value, so the equation has one root counted twice. Graphically the parabola touches the x-axis at a single point — its vertex — rather than crossing it.
How do I build a quadratic from its roots?
Use the sum and product: if the roots are r₁ and r₂, the equation is x² − (r₁ + r₂)x + (r₁ × r₂) = 0. For roots 4 and −1, that gives x² − 3x − 4 = 0.