The formula this page uses
x = ( −b ± √(b² − 4ac) ) / (2a)
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Quadratic Formula Calculator
Solve quadratic formula problems with clear steps, notation, and a final check.
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Calculate without using AI.
Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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General math
Quadratic Formula Calculator explained
The short version
- The quadratic formula solves every equation of the form ax² + bx + c = 0, even the ones that will not factor.
- The part under the square root, b² − 4ac, is called the discriminant, and its sign tells you the answer's shape before you finish.
- The whole top line, −b plus or minus the root, gets divided by 2a. That fraction bar covers everything.
What each part means
SymbolWhat it means
a — Coefficient of x²Must not be zero, or the equation is a straight line instead. Its sign says whether the parabola opens up or down.
b — Coefficient of xCopy its sign into the formula. A −5 in the equation becomes −(−5) = +5 on top.
c — Constant termThe number with no x. It is also the height where the parabola crosses the y-axis.
b² − 4ac — DiscriminantA single number that decides whether you get two roots, one root, or a complex pair.
Show your work: a full example
- The equation2x² + 5x − 12 = 0, so a = 2, b = 5, c = −12
- Discriminant first5² − 4(2)(−12) = 25 + 96 = 121
- Square-root it√121 = 11, a whole number, so the roots will be tidy
- Build the fractionx = (−5 ± 11) / (2 × 2) = (−5 ± 11) / 4
- Take the plus branch(−5 + 11) ÷ 4 = 6 ÷ 4 = 1.5
- Take the minus branch(−5 − 11) ÷ 4 = −16 ÷ 4 = −4
- Check both in the original2(1.5)² + 5(1.5) − 12 = 4.5 + 7.5 − 12 = 0 and 2(−4)² + 5(−4) − 12 = 32 − 20 − 12 = 0
A second, different case
- A different case: the discriminant goes negative2x² + 3x + 6 = 0, so a = 2, b = 3, c = 6
- Discriminant3² − 4(2)(6) = 9 − 48 = −39
- A negative under the root means no real answer√(−39) = i√39, where i is the square root of −1
- Build the fraction anywayx = (−3 ± i√39) / 4
- Put it in decimal form√39 ≈ 6.245, so x ≈ −0.75 + 1.561i and x ≈ −0.75 − 1.561i
- Read the graphthe parabola opens upward and its lowest point sits above the x-axis, so it never crosses
Copy-ready example
2x² + 5x − 12 = 0, so a = 2, b = 5, c = −12
The equation
Questions about the Quadratic Formula Calculator
Do I always have to use the quadratic formula?
No. Try factoring first, because 2x² + 5x − 12 splits neatly into (2x − 3)(x + 4) and takes ten seconds. Reach for the formula when nothing factors, when the coefficients are decimals, or when the discriminant turns out negative.
What can I learn from the discriminant before I finish?
Its sign alone answers most multiple-choice questions. Positive means the parabola cuts the x-axis twice, zero means it grazes it once, negative means it misses entirely. You often never need the roots themselves.
Why is the denominator 2a and not just a?
The formula comes from completing the square on ax² + bx + c. Halving b/a during that step is where the 2 appears, and it stays in the final denominator.
What happens if a = 0?
Then there is no x² term, the equation is linear, and 2a in the denominator becomes zero. Solve bx + c = 0 directly instead: x = −c/b.
A calculator gives you the number. If you also want to be able to do this without one, the quadratic formula explained walks through the method step by step, with a worked example and the mistakes that cost marks.