Quadratic Formula Calculator

How this quadratic formula calculator works

Any equation that can be written as ax² + bx + c = 0 is solved by the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. This quadratic formula calculator takes your three coefficients, computes the discriminant b² − 4ac first, and then substitutes everything into the formula exactly the way you would on paper, so the "Step-by-step" section matches what a teacher expects to see. When the coefficients are whole numbers it also gives the exact answer with a simplified square root, such as (3 ± √5) / 2, alongside the decimal.

Worked example: x² − 3x + 2 = 0

Here a = 1, b = −3 and c = 2. The discriminant is (−3)² − 4(1)(2) = 9 − 8 = 1. Because it is positive, there are two real roots. Substitute: x = (3 ± √1) / 2 = (3 ± 1) / 2, so x = 2 or x = 1. You can check by factoring: (x − 1)(x − 2) = x² − 3x + 2. The parabola has its vertex at x = −b / 2a = 1.5, y = −0.25, and opens upward because a is positive.

Now change c to 5: x² + 2x + 5 = 0 has discriminant 4 − 20 = −16. The square root of a negative number is not real, so we write √(−16) = 4i and the roots are x = (−2 ± 4i) / 2 = −1 ± 2i. The parabola floats above the x-axis and never crosses it, but the equation still has two solutions in the complex numbers.

What the discriminant tells you before you solve

The sign of b² − 4ac decides the whole character of the answer. Positive: two different real roots, the graph crosses the x-axis twice. Zero: one repeated root, the graph just touches the axis (a perfect square trinomial like x² + 2x + 1 = (x + 1)²). Negative: no real roots, two complex conjugates. If the discriminant is a perfect square, the roots are rational and the quadratic factors nicely; if it is not, expect a square root in the exact answer.

Common mistakes

• Dropping the sign of b. The formula starts with −b. For b = −3 that is −(−3) = +3. The calculator writes this substitution out so you can see it.
• Dividing only the square root by 2a. The whole numerator, −b ± √D, is divided by 2a, not just the root.
• Not moving everything to one side. 2x² = 5x − 3 must become 2x² − 5x + 3 = 0 before you read off a, b and c.
• Squaring a negative b wrong. (−3)² is 9, not −9. The discriminant uses b², which is never negative.

When a is zero

If a = 0 there is no x² term and the formula breaks (you would divide by zero). The equation is just bx + c = 0, which has one solution, x = −c / b. The calculator detects this and solves the linear equation instead. If both a and b are 0, the equation is either always true (c = 0) or never true (c ≠ 0), and it says which. If you would rather factor than use the formula, the factor calculator will list the factor pairs of your constant term so you can look for the pair that adds to b.