What does the Quadratic Formula Calculator solve?

It solves any equation in the form ax² + bx + c = 0, returning both values of x using x = (−b ± √(b² − 4ac)) / 2a, along with the discriminant and the vertex of the parabola.

What is the discriminant and why does it matter?

The discriminant is b² − 4ac. A positive value means two distinct real roots, zero means one repeated real root, and a negative value means the two roots are complex conjugates.

How are complex roots formatted?

When the discriminant is negative, the Quadratic Formula Calculator writes the pair as a+bi and a−bi, simplifying to just i or −i when the real part is zero and the imaginary coefficient is 1.

Why does the calculator reject a = 0?

If a is 0 the x² term disappears and the equation becomes linear (bx + c = 0), which is not a quadratic — the calculator shows an error instead of dividing by zero.

What is the vertex value shown alongside the roots?

It is the vertex (x, y) of the parabola y = ax² + bx + c, computed at x = −b / 2a. This is the parabola's minimum or maximum point, useful when sketching the graph.

Is my equation sent anywhere when I use this calculator?

No. The Quadratic Formula Calculator computes every result in your browser — the coefficients you enter are never uploaded to ArrayKit.

Quadratic Formula Calculator Online

Solve ax² + bx + c = 0 for real or complex roots with the discriminant and vertex shown step by step. Your numbers stay on your device.

The Quadratic Formula Calculator runs entirely in your browser. The coefficients you enter and the roots it computes never leave your device.

Open the Factor Calculator

About Quadratic Formula Calculator

The Quadratic Formula Calculator solves any equation of the form ax² + bx + c = 0 and returns both roots instantly, whether they are real or complex. Enter the three coefficients a, b, and c and it applies x = (−b ± √(b² − 4ac)) / 2a, showing the discriminant that determines the root type and the vertex of the parabola y = ax² + bx + c. When the discriminant is negative, the calculator formats the conjugate pair as a+bi and a−bi instead of showing an error. It is built for algebra students checking homework, teachers preparing worked examples, and anyone who needs a fast, reliable second root for a physics or engineering problem. Everything runs locally in your browser — the coefficients you type never leave your device.

Features

Solves ax² + bx + c = 0 for both roots in one step
Handles real, repeated, and complex conjugate roots automatically
Shows the discriminant b² − 4ac so you can see why the roots came out that way
Formats complex roots clearly as a+bi and a−bi
Calculates the vertex (x, y) of the parabola for graphing
Copy a plain-text summary of the equation and results
Rejects a = 0 with a clear message instead of a silent wrong answer
Runs entirely in your browser with no equation data sent anywhere

How to use the Quadratic Formula Calculator

Enter the coefficient a for the x² term
Enter the coefficient b for the x term
Enter the constant c
Read the discriminant, both roots, and the vertex, or copy the summary

Example

Input

x²-3x+2=0

Output

x = 2, x = 1

a=1, b=-3, c=2 factors to (x-2)(x-1), so the roots are 2 and 1.

Common errors & troubleshooting

Entering a = 0 gives an error instead of a root. — When a is 0 the equation is linear, not quadratic — the Quadratic Formula Calculator flags this because there is no ax² term to apply the formula to.
The result shows roots like 1+2i and 1−2i instead of numbers. — That means the discriminant is negative, so the equation has no real solutions — 1+2i and 1−2i are the correct complex conjugate roots.
Typing the equation as '3x2-5x+2' with no operators confuses the fields. — The calculator expects one coefficient per box (a, b, c) — enter 3, -5, and 2 separately rather than the whole equation as text.
A negative b or c is ignored. — Include the minus sign directly in the field, e.g. type -3 for b, not 3 with a separate subtraction — the calculator reads the sign as part of the number.

Frequently asked questions

What does the Quadratic Formula Calculator solve?: It solves any equation in the form ax² + bx + c = 0, returning both values of x using x = (−b ± √(b² − 4ac)) / 2a, along with the discriminant and the vertex of the parabola.
What is the discriminant and why does it matter?: The discriminant is b² − 4ac. A positive value means two distinct real roots, zero means one repeated real root, and a negative value means the two roots are complex conjugates.
How are complex roots formatted?: When the discriminant is negative, the Quadratic Formula Calculator writes the pair as a+bi and a−bi, simplifying to just i or −i when the real part is zero and the imaginary coefficient is 1.
Why does the calculator reject a = 0?: If a is 0 the x² term disappears and the equation becomes linear (bx + c = 0), which is not a quadratic — the calculator shows an error instead of dividing by zero.
What is the vertex value shown alongside the roots?: It is the vertex (x, y) of the parabola y = ax² + bx + c, computed at x = −b / 2a. This is the parabola's minimum or maximum point, useful when sketching the graph.
Is my equation sent anywhere when I use this calculator?: No. The Quadratic Formula Calculator computes every result in your browser — the coefficients you enter are never uploaded to ArrayKit.

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