Free Quadratic Formula Calculator

Stuck on ax² + bx + c = 0? Punch in a, b and c below and this quadratic formula calculator spits out both roots real numbers if there are any, complex numbers if there aren't along with every step it took to get there.

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Quadratic Formula Calculator

ax2 + bx + c = 0

a can't be 0, otherwise there's no x² term and it isn't a quadratic anymore.

What the Discriminant Tells You

b² − 4ac = 0
one repeated real root
b² − 4ac > 0
two different real roots
b² − 4ac < 0
two complex roots

Answer

Solution:

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Opens with a, b, c and this result already filled in.

Enter a, b and c and both roots, worked out step by step, will appear here.

How to Use This Calculator

Three boxes. That's really all there is to it.

  1. Type your a, b and c values from ax² + bx + c = 0 into the matching boxes.
  2. Hit Calculate. You'll get both roots straight away the calculator figures out on its own whether they're real numbers or complex ones and formats them accordingly.
  3. Scroll down a little and you'll see the whole thing worked out, step by step, exactly the way you'd do it by hand.
  4. Copy the answer if you just need the numbers, print a clean version with just your equation and the working (no ads, no sidebar), or share a link that reopens with your exact values and result already there.
  5. Clear resets a, b and c if you want to try a different equation.

The Formula, Worked Through

x = (−b ± √(b² − 4ac)) ÷ 2a

That little symbol under the square root, b² − 4ac, is called the discriminant, and it's basically deciding your fate before you even finish the calculation. Positive, and you get two real answers. Zero, and both answers collapse into the same number. Negative, and you're headed into complex number territory.

Take 12x² + 28x + 15 = 0. Here a=12, b=28, c=15.

b² − 4ac = 28² − 4(12)(15) = 784 − 720 = 64
√64 = 8
x = (−28 ± 8) ÷ 24

x = −20/24 = −0.833333
x = −36/24 = −1.5

Discriminant came out positive, so two clean real answers, no surprises there.

Why Not Just Do It by Hand?

You absolutely can, and for something like x² − 5x + 6 = 0 you probably should, it's quick. But a few things get messy fast:

  • Square roots of negative numbers trip people up constantly. Miss that the discriminant went negative and you'll end up trying to take the square root of a negative number the "normal" way, which just doesn't work.
  • Simplifying the radical by hand (turning √44 into 2√11, say) is a step people skip or get wrong under time pressure.
  • Bigger coefficients mean more room for a sign error somewhere in the middle, and one small slip changes the whole answer.
  • Seeing the full working means you can actually check where you went wrong, instead of just knowing your answer doesn't match.

What a, b and c Actually Mean

  • a - the coefficient on x². Can't be zero, or the equation stops being quadratic at all.
  • b - the coefficient on x. Can be positive, negative, or zero.
  • c - the constant term, with no x attached. Also fine as zero (in fact, if c is 0, x=0 is always one of your answers).
  • The discriminant (b²−4ac) - not something you type in, but it's what the formula quietly checks first to decide whether your answers will be real or complex.

Frequently Asked Questions

That's a complex number, and it shows up whenever the discriminant works out negative. i just stands for the square root of negative 1, which doesn't exist among ordinary real numbers, so mathematicians invented a symbol for it instead of just giving up.

That plus-or-minus sign in the formula splits into two separate calculations, one with plus and one with minus, which is exactly why quadratics generally have two solutions instead of one. If the discriminant is exactly zero those two answers happen to land on the same number.

b and c can be anything, including negative, zero, or a decimal. Only a has a real restriction, which is that it can't be zero, since that would remove the x squared term entirely and the equation wouldn't be quadratic anymore.

You'll need to rearrange things first so everything sits on one side and the equation reads ax² + bx + c = 0. Once it's in that shape, pull out a, b and c and you're good to go.

No, it only carries the exact a, b and c you typed in, nothing else.

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