Quadratic equation solver (with steps)
Enter the coefficients a, b and c and this solver works through the quadratic formula the way an examiner expects: discriminant first, then exact roots in surd form, then decimals rounded sensibly. It also gives the completed-square form and the turning point, which is what most GCSE and A Level questions actually ask for after the roots.
Enter ax² + bx + c = 0
Discriminant: b² − 4ac = 25 — two distinct real roots.
Exact roots: 4 and -1
x = 4 or x = -1
Completed square: (x − 1.5)² − 6.25
Turning point: (1.5, -6.25)
The method behind it
- Discriminant: Δ = b² − 4ac decides how many real roots exist.
- Quadratic formula: x = (−b ± √Δ) / 2a.
- Completed square: a(x + b/2a)² + (c − b²/4a), giving the turning point directly.
How to use this tool
1. Rearrange to standard form
Write your equation as ax² + bx + c = 0 with everything on one side, so the constant term is on the left.
2. Enter the coefficients
Type a, b and c into the boxes. Negative values and decimals are fine; a cannot be zero.
3. Read the working
Check the discriminant to see how many real roots exist, then use the exact surd roots for non-calculator papers and the decimals for calculator papers.
Frequently asked questions
How do you solve a quadratic equation?
Write it as ax² + bx + c = 0, then either factorise, complete the square, or use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The formula always works, so it is the safest method under exam pressure when factorising is not obvious.
What does the discriminant tell you?
The discriminant is b² − 4ac. If it is positive there are two distinct real roots, if it is zero there is one repeated root, and if it is negative there are no real roots (the curve never crosses the x-axis).
Do I need to leave answers in surd form?
On non-calculator papers, yes — an exact answer such as (3 ± √5)/2 earns the accuracy mark, while a rounded decimal may not. On calculator papers give the decimal to the accuracy the question asks for, usually 2 or 3 significant figures.
How do I find the turning point of a quadratic?
Complete the square to get a(x + p)² + q; the turning point is then (−p, q). This solver shows the completed-square form and the turning point for the coefficients you enter.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.