Quadratic Equation Solver

Enter the coefficients of ax² + bx + c = 0. Sumzo finds the exact roots, shows each step, and checks every root by substituting it back into the equation.

Whole numbers, decimals or fractions such as 3/4
Used for decimal approximations

Result

Enter values and press Calculate to see the result.

Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The quadratic formula gives both roots of ax² + bx + c = 0 whenever a ≠ 0.

Δ=b2−4ac\Delta = b^2 - 4ac

The discriminant Δ tells you what kind of roots to expect: two distinct real roots if Δ > 0, one repeated real root if Δ = 0, and a pair of complex conjugate roots if Δ < 0.

How to use this calculator

  1. Rearrange your equation so that one side is 0. For example, 2x² = 4x + 6 becomes 2x² − 4x − 6 = 0.
  2. Enter a, b and c. Use 0 for a missing term: x² − 9 = 0 has b = 0.
  3. Press Calculate. If you change an input afterwards, the result is marked as outdated until you calculate again.
  4. Read the exact roots first. Decimal approximations appear beneath them and are labelled as approximate.

Worked example

Solve 2x² − 4x − 6 = 0

Here a = 2, b = −4 and c = −6.

The discriminant is Δ = (−4)² − 4 · 2 · (−6) = 16 + 48 = 64. It is positive, so there are two distinct real roots.

Apply the formula: x = (4 ± √64) / 4 = (4 ± 8) / 4.

So x = 12 / 4 = 3 or x = −4 / 4 = −1.

Check x = 3: 2 · 3² − 4 · 3 − 6 = 18 − 12 − 6 = 0.

Frequently asked questions

What happens if a = 0?

Then the equation is not quadratic — it is the linear equation bx + c = 0. Sumzo tells you this and solves it as a linear equation (x = −c/b), or explains that there is no solution or that every number is a solution.

Why do some answers contain √ or i?

Sumzo keeps roots exact. When the discriminant is not a perfect square, the roots are irrational and are written with a square root, such as (−1 + √5)/2. When the discriminant is negative there are no real roots, and the roots are complex numbers written with i, where i² = −1. A rounded decimal is shown alongside each exact root.

What does “Verified” mean on this calculator?

After solving, Sumzo substitutes each exact root back into ax² + bx + c and confirms the result is exactly 0, and checks that the sum and product of the roots equal −b/a and c/a. These checks use exact arithmetic, not rounded decimals.

Can I enter fractions or decimals?

Yes. Enter values such as 0.5, −3 or 3/4. Use a period as the decimal point and leave out thousands separators.

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