Quadratic Equation Solver

What this calculator does

This tool solves any quadratic equation of the form ax² + bx + c = 0. You enter the three coefficients and it instantly returns the roots, the discriminant, the vertex of the parabola, and the axis of symmetry. When the discriminant is negative it reports the complex conjugate roots rather than failing, and when the leading coefficient is zero it gracefully solves the equation as a linear one. Every value recalculates as you type, and all of the arithmetic happens locally in your browser.

Why you might need it

Quadratic equations are one of the first non-trivial equations students learn to solve, and they appear far beyond the classroom. Physics uses them for projectile motion — the time a thrown object spends in the air is the root of a quadratic. Engineering and finance use them in optimisation problems where the maximum or minimum of a parabola matters. A solver that shows the discriminant, the vertex, and the axis of symmetry alongside the roots is useful both as a homework check and as a quick reference when you only need the answer and the shape of the curve.

How to use it

1. Enter the coefficient a (the x² term), b (the x term), and c (the constant).
2. Read the roots in the result card — they update as you type.
3. Check the discriminant, vertex, and axis of symmetry cards for the full picture of the parabola.
4. Use the copy button to grab the roots, and Reset to return to the example equation x² − 3x + 2 = 0.

How it’s calculated

The roots come from the quadratic formula:

x = (−b ± √(b² − 4ac)) ÷ 2a

The expression under the square root, Δ = b² − 4ac, is the discriminant. When Δ is positive the calculator takes its real square root and produces two roots. When Δ is zero both roots collapse to the single value −b/2a. When Δ is negative the calculator takes the square root of −Δ and writes the answer as the complex conjugate pair −b/2a ± (√(−Δ)/2a)i.

The vertex of the parabola sits at x = −b/2a; substituting that x back into ax² + bx + c gives the y-coordinate. The axis of symmetry is the vertical line through the vertex, x = −b/2a. If a is zero the equation is not quadratic at all, so the tool solves the linear form bx + c = 0 as x = −c/b.

Common pitfalls

The most common error is sign mistakes when reading off coefficients — remember that in x² − 3x + 2 the value of b is −3, not 3. Another is forgetting that a negative discriminant does not mean “no answer”; it means the answers are complex. People also sometimes mix up the vertex x-value (−b/2a) with a root; the vertex is the turning point and only coincides with a root when the discriminant is zero. Finally, dividing by a is invalid when a is zero, which is why the linear case must be handled separately.

Tips and related calculations

If your equation is given in vertex form, a(x − h)² + k, you can expand it to standard form before entering the coefficients here. The two real roots always sit symmetrically around the axis of symmetry, so their average equals −b/2a — a quick sanity check. The product of the roots equals c/a and their sum equals −b/a, another pair of relationships worth remembering. Because the whole calculation runs on your device, you can experiment with coefficients freely and watch the discriminant flip between positive, zero, and negative.