Quadratic Formula Calculator
Solve ax² + bx + c = 0 for real or complex roots: with the discriminant, vertex, and every step shown.
Details
Solutions for x
x = 2 or x = 1
x² − 3x + 2 = 0, two distinct real roots (the parabola crosses the x-axis twice)
The two roots added
3
The two roots multiplied
2
This solves any quadratic equation. Enter a, b and c from ax² + bx + c = 0 and it returns both roots.
It also gives the discriminant, which tells you whether to expect two answers, one or none, plus the turning point of the curve and the factored form where the roots are tidy enough for one to exist.
What is a quadratic equation?
A quadratic is any equation where the highest power is x squared. Written in standard form it looks like ax² + bx + c = 0.
Solving it means finding the values of x that make the equation true. Those values are called the roots, and a quadratic usually has two of them.
Drawn on a graph, a quadratic makes a U-shaped curve called a parabola. The roots are wherever that curve crosses the horizontal axis, which is why there are normally two: the curve goes down through the line and comes back up through it.
Take the signs with the numbers. In x² − 5x + 6, a is 1, b is −5 and c is 6. Forgetting that b is negative is the most common slip in the whole topic.
What to enter
- a (x² coefficient)
- The number multiplying x². It cannot be zero, because without an x² term the equation is not quadratic.
- b (x coefficient)
- The number multiplying x. Include the minus sign if there is one.
- c (constant)
- The number with no x attached.
What this assumes
Your equation is arranged with everything on one side and zero on the other. Rearrange first if it is not.
When there is no real answer, the results are shown as complex numbers using i.
How to calculate the quadratic formula
One formula solves every quadratic, whether or not it factorises neatly.
- ±
- Do it twice: once adding, once subtracting. That gives the two roots
- b² − 4ac
- The discriminant. It decides how many answers exist
Write down a, b and c. Take the signs with them. Getting this wrong is what causes most wrong answers.
Work out the discriminant. Calculate b² − 4ac first, on its own. If it is positive there are two answers, if zero there is one, if negative there are none among ordinary numbers.
Take its square root. Then do −b plus that, and −b minus that. The square root calculator will simplify an untidy one.
Divide both by 2a. Both, not just the last part. Forgetting to divide the whole top is the other classic mistake.
See a worked example: x² − 5x + 6 = 0
- a
- 1
- b
- −5
- c
- 6
Discriminant: b² − 4ac = 25 − 24 = 1. It is positive, so expect two answers.
Square root of 1 is 1.
−b is +5. So the two tops are 5 + 1 = 6 and 5 − 1 = 4.
Divide each by 2a, which is 2: 6 ÷ 2 = 3, and 4 ÷ 2 = 2.
Check by multiplying back: (x − 2)(x − 3) = x² − 5x + 6. Correct.
x = 3 and x = 2
Frequently asked questions
It is b² − 4ac, and it tells you how many answers exist before you do the rest of the work.
Positive means two different answers, so the curve crosses the axis twice. Zero means one repeated answer, so the curve just touches the axis. Negative means no real answers, so the curve never reaches the axis at all.
Because the curve is a U shape, so it normally crosses the horizontal axis in two places, once on the way down and once on the way back up.
Both are genuine solutions to the equation. In a real-world problem, though, one is often impossible, such as a negative length or a time before the start, so you keep the one that makes sense.
Try factorising first if a is 1 and you can spot two numbers that multiply to c and add to b. For x² − 5x + 6 those are −2 and −3, giving the answers in seconds.
If nothing obvious appears within about twenty seconds, switch to the formula. It always works, including for answers with decimals or surds.
It means the discriminant was negative, so there is no ordinary number whose square is negative. Mathematicians handle this with i, defined as the square root of −1.
On a graph it means the curve never touches the horizontal axis. If your question came from a real situation, that usually means the thing being asked about never actually happens.
Problems people actually run into
Losing the minus sign on b
In x² − 5x + 6, b is −5, not 5. The formula then needs −b, which is +5. Two negatives that cancel is exactly where people slip.
Write a, b and c down explicitly with their signs before touching the formula. It costs five seconds and prevents most errors in this topic.
Dividing only part of the top by 2a
The whole of −b ± √(b² − 4ac) is divided by 2a, not just the square root part.
It is worth writing the fraction out with a proper horizontal line rather than a slash, because a slash makes it easy to forget what the division applies to.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: August 29, 2026