Enter the coefficients a, b and c of ax² + bx + c = 0 and get the roots — whether they’re two real numbers, one repeated root, or a complex conjugate pair. The tool also shows the discriminant and the parabola’s vertex. It computes with the quadratic formula and runs on your device.
Enter the coefficients of ax² + bx + c = 0
Enter a, b and c to solve.
How it works
It uses the quadratic formula, x = (−b ± √(b²−4ac)) / 2a. The part under the root, b²−4ac, is the discriminant: positive gives two real roots, zero gives one repeated root, and negative gives a complex conjugate pair (real ± imaginary). The tool picks the right case automatically.
The vertex — the turning point of the parabola — is at x = −b/2a, with its y found by substituting back. Coefficient a must not be zero; if it is, the equation is linear, not quadratic, and the tool says so rather than dividing by zero.
Practical examples
Two real roots
For x² − 5x + 6 = 0 (a=1, b=−5, c=6) the discriminant is 1 and the roots are x = 2 and x = 3 — the equation factors as (x−2)(x−3).
A repeated root
x² − 4x + 4 = 0 has discriminant 0 and a single root x = 2 (a perfect square). The vertex sits exactly on the x-axis at (2, 0).
Complex roots
x² + 1 = 0 has discriminant −4 and no real solution; the roots are 0 + 1i and 0 − 1i, a conjugate pair the tool shows in a ± bi form.
Frequently asked questions
What does the discriminant tell me?
The sign of b²−4ac reveals the nature of the roots before you solve: positive means two distinct real roots, zero means one repeated real root, and negative means two complex conjugate roots. Its size also reflects how far apart the real roots are.
What if there’s no real solution?
When the discriminant is negative the parabola never crosses the x-axis, so there are no real roots. The tool still solves it over the complex numbers, giving a conjugate pair written as real ± imaginary (a ± bi).
Why must a not be zero?
If a is 0 the x² term vanishes and the equation is linear (bx + c = 0), not quadratic. Dividing by 2a would be dividing by zero, so the tool flags a = 0 and asks for a non-zero leading coefficient.
Can I use decimal or negative coefficients?
Yes. Enter any real numbers, including negatives and decimals — a comma or dot both work for the decimal point. For example a = 0.5, b = 1, c = −4 is perfectly valid.
What is the vertex used for?
The vertex is the parabola’s highest or lowest point. It gives the minimum or maximum value of the expression and the axis of symmetry (x = −b/2a), which is useful in optimisation and graphing problems.
How precise are the roots?
They’re computed at full floating-point precision and shown to four decimals. For most homework and engineering use that’s ample; irrational roots like √2 appear as their decimal approximation.
Does it show the factored form?
Not directly, but the roots give it to you: if the roots are r₁ and r₂, the equation factors as a(x − r₁)(x − r₂). With a repeated root r it’s a(x − r)². The tool focuses on the roots, discriminant and vertex.
Are my coefficients kept private?
Yes. The solving happens entirely in your browser; nothing is uploaded and analytics never receives the numbers.
Related tools
GCD and LCM Calculator
Find the greatest common divisor and least common multiple of two or more whole numbers. Runs in your browser.
Math & education
Prime Number Checker
Check whether a number is prime, with its prime factorisation and the nearest primes. Runs in your browser.
Math & education
Number Base Converter
Convert whole numbers between binary, octal, decimal and hexadecimal at once. BigInt-precise, in your browser.
Math & education
Roman Numeral Converter
Convert Roman numerals to numbers and back, for values from 1 to 3999. Auto-detects the direction. Runs in your browser.
Math & education