Quadratic Equation Calculator

What is a quadratic equation?

A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a, b and c are known numbers and a is not zero. The highest power of x is 2, which gives the equation its name (quadratus is Latin for square). Quadratics appear wherever one quantity depends on the square of another: the height of a thrown ball over time, braking distance against speed, the area of a garden with a fixed length of fence, or the profit of a product whose margin shrinks as volume grows. Solving the equation means finding the values of x, called the roots or solutions, that make the left-hand side equal to zero.

The quadratic formula

Every quadratic equation, including those that do not factor neatly, is solved by:

x = (−b ± √(b² − 4ac)) ÷ 2a

The ± sign produces two values, one using plus and one using minus. The expression under the square root, b² − 4ac, is called the discriminant and decides what kind of roots you get.

Example: solve x² − 3x + 2 = 0, so a = 1, b = −3 and c = 2.

1. Discriminant: D = (−3)² − 4 × 1 × 2 = 9 − 8 = 1.
2. Square root: √1 = 1.
3. Formula: x = (3 ± 1) ÷ 2, so x₁ = 4 ÷ 2 = 2 and x₂ = 2 ÷ 2 = 1.
4. Check: 2² − 3 × 2 + 2 = 4 − 6 + 2 = 0 and 1² − 3 × 1 + 2 = 0. Both work.

Because the roots are 2 and 1, the quadratic factors as (x − 2)(x − 1) = 0. The sum of the roots is 3, which equals −b ÷ a, and the product is 2, which equals c ÷ a. These two relationships (Vieta's formulas) are a quick way to check any answer.

What the discriminant tells you

Completing the square

The quadratic formula is what you get by completing the square on the general equation, and the method is worth knowing on its own. For x² − 3x + 2 = 0: move the constant across (x² − 3x = −2), add the square of half the x-coefficient to both sides ((3 ÷ 2)² = 2.25, giving x² − 3x + 2.25 = 0.25), rewrite the left side as a perfect square ((x − 1.5)² = 0.25), take the square root of both sides (x − 1.5 = ±0.5) and solve: x = 2 or x = 1. The intermediate form a(x − h)² + k is called vertex form, because (h, k) is the vertex of the parabola.

What the graph shows

The graph of y = ax² + bx + c is a parabola. It opens upward when a is positive and downward when a is negative. The real roots are the points where the curve crosses the x-axis, and the constant c is where it crosses the y-axis. The vertex sits on the axis of symmetry x = −b ÷ 2a and has a y-coordinate of c − b² ÷ 4a. For the example, the vertex is (1.5, −0.25): the curve dips just below the axis between the two roots, which is why there are two of them. If you need the gradient of a straight line instead, see the slope calculator.

How to use this calculator

1. Rearrange your equation into standard form with zero on the right-hand side. 2x² = 5x − 3 becomes 2x² − 5x + 3 = 0.
2. Type a, b and c with their signs. Use 0 for a missing term: x² − 9 = 0 has a = 1, b = 0, c = −9.
3. Check the rendered equation under the inputs matches what you meant to solve.
4. Read the roots, the discriminant and the vertex, and follow the numbered steps to see the working.

Common mistakes

• Sign errors with b: when b = −3, −b is +3, and b² is +9, never −9. Squaring a negative gives a positive.
• Forgetting the ±: a quadratic normally has two roots. Writing only the "+" version loses one of them.
• Dividing only part of the top: the whole of −b ± √D is divided by 2a, not just the square root.
• Reading coefficients out of order: in 5 + x² = 4x, a = 1, b = −4 and c = 5 once it is in standard form.
• Rounding too early: keep √D exact until the last step. The calculator shows up to 4 decimal places, so irrational roots are rounded.
• Fractional coefficients: clear them first with the fraction calculator or multiply through by the common denominator; powers and roots are covered by the exponent calculator.