Quadratic Equation Solver
Step-by-Step Working ⚡
Enter the coefficients a, b, and c for any quadratic equation in the form ax² + bx + c = 0 and get instant roots with full working shown.
ax² + bx + c = 0
Quick Examples
The Quadratic Formula
For any equation ax² + bx + c = 0, the roots are:
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant (b² − 4ac) determines the nature of the roots:
- Positive → two distinct real roots
- Zero → one repeated real root
- Negative → two complex (imaginary) roots
Related Calculators
Frequently Asked Questions
What if a = 0?
If a = 0, the equation is linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0.
What are complex roots?
When the discriminant is negative, the square root of a negative number produces imaginary roots in the form p ± qi, where i = √−1. These appear in pairs and are used in advanced maths, physics, and engineering.
What does the discriminant tell us?
The discriminant b² − 4ac tells you how many real solutions exist without fully solving the equation. It is frequently tested in GCSE and A-level exams.
Can I factorise instead?
For simple integer roots, factorising is quicker. The quadratic formula always works regardless of whether the equation factorises neatly.