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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