What is the quadratic formula?

For ax² + bx + c = 0 with a not equal to 0, the roots are x = (−b ± √(b² − 4ac)) ÷ 2a. The ± gives the two solutions, and the expression under the square root, b² − 4ac, is the discriminant.

What does the discriminant tell me?

The discriminant Δ = b² − 4ac decides the nature of the roots. If Δ is greater than 0 there are two distinct real roots; if Δ equals 0 there is one repeated real root (a double root); if Δ is less than 0 there are two complex conjugate roots and the parabola never crosses the x-axis.

How does the solver handle complex roots?

When the discriminant is negative the solver takes the square root of its absolute value and writes the answer in the form p ± qi, where i is the imaginary unit (i² = −1). The two roots are always conjugates of each other, sharing the same real part and opposite imaginary parts.

How are the vertex and axis of symmetry found?

The axis of symmetry is the vertical line x = −b ÷ 2a, which is also the x-coordinate of the vertex. Substituting that x back into f(x) = ax² + bx + c gives the vertex y-value, so the vertex is the turning point (minimum if a is positive, maximum if a is negative).

If a = 0 the equation is no longer quadratic. With b not zero it becomes the linear equation bx + c = 0, whose single root is x = −c ÷ b. If both a and b are 0 the solver reports either an identity (when c = 0) or no solution (when c is non-zero).

Yes. The solver runs entirely in your browser using ordinary arithmetic — no coefficients or results are ever sent to a server.

Quadratic Equation Solver

A quadratic equation solver that takes the three coefficients of ax² + bx + c = 0 and returns everything you need: the two roots (real or complex), the discriminant, the vertex and axis of symmetry of the parabola, and a numbered, copyable derivation that shows exactly how the answer was reached. It is built for students checking homework, teachers preparing worked examples, and anyone who needs a fast, reliable solution without installing software or sending data anywhere.

How it works

A quadratic is any equation that can be written in the standard form ax² + bx + c = 0, where a is not zero. The solver applies the quadratic formula:

x = (−b ± √(b² − 4ac)) ⁄ 2a

The quantity under the square root, b² − 4ac, is the discriminant and it controls the whole result. When the discriminant is positive the square root is a real number and the ± produces two distinct real roots. When it is exactly zero the ± collapses to a single value, giving one repeated real root that sits right on the vertex. When the discriminant is negative the square root is imaginary, so the solver writes the two roots as a complex conjugate pair p ± qi.

Alongside the roots, the tool computes the axis of symmetry at x = −b ⁄ 2a and substitutes that value back into the function to find the vertex — the turning point of the parabola. If you set a to zero the equation degenerates to a linear one, bx + c = 0, and the solver handles that case too, reporting the single root x = −c ⁄ b (or flagging an identity or no-solution when b is also zero).

Worked example

Take x² − 3x + 2 = 0, so a = 1, b = −3, c = 2. The discriminant is Δ = (−3)² − 4(1)(2) = 9 − 8 = 1, which is positive, so expect two real roots. The square root of 1 is 1, and the formula gives x = (3 ± 1) ⁄ 2, that is x = 2 and x = 1. The axis of symmetry is x = 3 ⁄ 2 = 1.5, and the vertex is (1.5, −0.25) — the lowest point of an upward-opening parabola. You can verify the roots by factoring: (x − 1)(x − 2) = 0.

Now try x² + 1 = 0, where a = 1, b = 0, c = 1. The discriminant is 0 − 4 = −4, which is negative, so the roots are complex: x = ±i. The solver reports them as 0 + 1i and 0 − 1i, a perfect conjugate pair.

Equation	Discriminant	Roots
x² − 3x + 2 = 0	1	2 and 1
x² − 4x + 4 = 0	0	2 (double root)
x² + 1 = 0	−4	0 ± 1i
2x² + 4x − 6 = 0	64	1 and −3

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