Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 for its roots.
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About this calculator
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation written in the standard form ax² + bx + c = 0, where x is the unknown and a, b and c are known numbers with a ≠ 0 (if a were 0 the equation would be linear). Quadratics describe the arc of a thrown ball, the shape of a satellite dish, profit curves and countless other situations. They can be solved by factoring, completing the square, graphing or the quadratic formula — the one method that always works. This calculator applies the formula and reports every solution.
The Quadratic Formula
x = [ −b ± √(b² − 4ac) ] ÷ 2a
The ± means there are two answers: one using the plus sign and one using the minus. These values are the roots (or zeros) of the equation — the x-values where the parabola y = ax² + bx + c crosses the x-axis. The formula comes from completing the square on the general equation, and it also reveals the parabola's axis of symmetry, x = −b ÷ 2a.
The Discriminant Tells You What to Expect
The expression under the square root, D = b² − 4ac, is called the discriminant:
- D > 0: two distinct real roots — the parabola crosses the x-axis twice.
- D = 0: one repeated real root — the parabola just touches the axis at its vertex.
- D < 0: no real roots; there are two complex (imaginary) roots, and the parabola never reaches the axis.
A Quick Example
To solve x² − 5x + 6 = 0, use a = 1, b = −5, c = 6. The discriminant is 25 − 24 = 1, so x = (5 ± 1) ÷ 2, giving x = 3 and x = 2. You can check by factoring: (x − 2)(x − 3) = 0.
Tips for Solving Quadratics
- Move every term to one side so the equation equals zero before reading off a, b and c.
- Watch the signs — a negative b appears as −b in the formula, which turns it positive.
- Use the vertex, x = −b ÷ 2a, to find the maximum or minimum of the parabola.
For the square roots inside the formula, the Root Calculator is handy, and the Exponent Calculator covers the squares.
Worked example
For x² − 3x − 4 = 0, the discriminant is 25 (positive, so two real roots): x = 4 and x = −1.
Frequently asked questions
What does the discriminant tell you?
Its sign predicts the type of roots before you even solve: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots.
What are complex roots?
When the discriminant is negative, the square root of a negative number isn't a real number — the roots are expressed using i (the imaginary unit, where i² = −1) instead.
Does this work if a = 0?
No — with a = 0 the x² term disappears and the equation becomes linear (bx + c = 0), which has a different, simpler solution than the quadratic formula.
How do I check a root is correct?
Substitute it back into ax² + bx + c — a correct root makes the expression equal exactly 0.
