Quadratic Formula Calculator
Solve ax² + bx + c = 0 — real or complex roots, the discriminant, and the vertex.
The quadratic formula solves ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) / 2a. The part under the root, b² − 4ac, is the discriminant: positive gives two real roots, zero gives one, and negative gives a pair of complex roots. For x² − 3x + 2 = 0 the roots are 2 and 1. Enter a, b, and c to solve any quadratic.
Results are estimates provided for general information.
ax² + bx + c = 0
x₁ =2
x₂ =1
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About the Quadratic Formula Calculator
A quadratic equation has the form ax² + bx + c = 0, and the quadratic formula solves every one of them: x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a. This calculator applies it to the a, b, and c you enter and shows the full result, not just a number. The most telling piece is the discriminant — the b² − 4ac under the square root — because its sign decides everything. A positive discriminant means the parabola crosses the x-axis twice, so there are two distinct real roots. A discriminant of exactly zero means it just touches the axis, giving one repeated root. A negative discriminant means it never touches the axis, and the two roots are complex — a conjugate pair written a ± bi. The tool reports which of these cases you're in, gives the roots in the right form for that case, and also returns the vertex of the parabola, the turning point that's useful for graphing and for finding a maximum or minimum. If you enter a = 0 the equation isn't quadratic at all (there's no x² term), and the calculator says so rather than dividing by zero.
Frequently asked questions
What is the quadratic formula?+
For any equation ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / 2a. The ± means you work it out twice, once with plus and once with minus, which gives the two roots.
What does the discriminant tell you?+
The discriminant is b² − 4ac, the part under the square root. If it's positive there are two real roots; if it's zero there's one repeated real root; if it's negative there are two complex roots. It tells you the nature of the solutions before you finish solving.
What are complex roots?+
When the discriminant is negative, the square root of a negative number produces an imaginary part, so the roots take the form a ± bi where i is √(−1). They come as a conjugate pair. This calculator writes them out for you rather than erroring.
What is the vertex of a quadratic?+
The vertex is the turning point of the parabola — its lowest point when it opens upward, or highest when it opens downward. Its x-coordinate is −b / 2a. The vertex is handy for graphing and for finding a maximum or minimum value.
Why can't a be zero?+
If a = 0 there's no x² term, so the equation is linear (bx + c = 0), not quadratic, and the formula's division by 2a would be division by zero. The calculator flags this instead of producing a bad answer.
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