Free Quadratic Equation Solver
Paste any quadratic ax² + bx + c = 0 and see the full step-by-step working — factoring, completing the square or the quadratic formula, whichever fits the problem.
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Learn
The quadratic formula in action — worked step by step
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMOx^2 - 4x - 1 = 0
- 1
Try factoring first
Look for two integers that multiply to −1 and add to −4. None exist, so factoring is out — use the quadratic formula instead.
- 2
Identify a, b, c
In standard form ax² + bx + c = 0: a = 1, b = −4, c = −1.
- 3
Compute the discriminant Δ = b² − 4ac
Δ > 0 → two distinct real roots (irrational, since 20 isn't a perfect square).
- 4
Apply the quadratic formula
- 5
Simplify the surd
Final Answer
How to solve a quadratic equation — three methods that work
A quadratic equation is any equation that can be written in standard form ax² + bx + c = 0 with a ≠ 0. Three techniques cover every case students meet in Class 10, GCSE, A-Level and SAT maths. Factoring is fastest when a, b and c have clean integer factors. Completing the square is the go-to when the leading coefficient is 1 and factoring fails. The quadratic formula x = (−b ± √(b² − 4ac)) / 2a works for every quadratic, including ones with irrational or complex roots. The discriminant b² − 4ac tells you the nature of the roots before you even solve: positive means two real roots, zero means one repeated root, negative means two complex roots. Most textbook problems are designed around these three cases, so recognising which is which saves time in the exam.
Quadratic equations to practise (Class 10, GCSE, SAT)
Tap any problem to solve it with full step-by-step working.
- Solve with AI →
1. x^2 + 7x + 12 = 0
FactoringClass 10Easy - Solve with AI →
2. 2x^2 - 5x - 3 = 0
Quadratic formulaClass 10 / GCSEMedium - Solve with AI →
3. x^2 - 6x + 10 = 0
Complex rootsClass 11 / A-LevelMedium - Solve with AI →
4. Solve by completing the square: x^2 + 8x + 5 = 0
Completing the squareClass 10Medium - Solve with AI →
5. A rectangular garden has area 48 m² and perimeter 28 m. Find its dimensions.
Word problemClass 10 / GCSEHard
Frequently asked questions
What is the quadratic formula and when should I use it?+
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves every quadratic ax² + bx + c = 0. Use it whenever factoring by inspection isn't obvious, or when the roots are irrational or complex — two cases where factoring will fail.
How do I know whether a quadratic has real or complex roots?+
Compute the discriminant Δ = b² − 4ac. If Δ > 0 you get two distinct real roots; if Δ = 0 you get one repeated real root; if Δ < 0 you get a complex conjugate pair (no real solutions).
When is completing the square better than the quadratic formula?+
Completing the square is faster when the leading coefficient a = 1 and b is even, because you avoid the surd simplification step. It's also the technique A-Level and JEE papers use to derive vertex form y = a(x − h)² + k for graphing.
Can SolveGini solve quadratic word problems from CBSE and GCSE textbooks?+
Yes. Paste or snap the problem — the solver picks out the variables, sets up the quadratic, solves it, then explains each step in plain English. Word problems about areas, speeds, ages and projectile motion are all in scope.
Is the quadratic equation solver free to use?+
SolveGini has a free plan. Guests can try 1 solve per day with no signup. A free account adds 5 daily solves, quizzes, flashcards and the study planner. Step-by-step working is never paywalled — premium only raises the daily limit.
Does it cover JEE-level quadratic topics like sum and product of roots?+
Yes. Sum α + β = −b/a, product αβ = c/a, and the usual JEE questions on α² + β² or symmetric-function identities are handled step by step. It also solves quadratic inequalities and problems where the roots must satisfy an additional condition.
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