Quadratic Formula Calculator
This quadratic formula calculator solves any equation shaped like ax² + bx + c = 0. Type the three numbers and you'll get both roots, the discriminant, and the vertex, plus a graph of the parabola so you can see where it crosses the x-axis. It handles real and complex answers, and it shows every step of the formula, so it's as good for homework as it is for a quick solve.
- Both roots
- Real and complex
- Discriminant and vertex
- Live parabola graph
- Steps shown
Last updated June 16, 2026 x = (−b ± √(b² − 4ac)) ÷ 2a Reviewed by the Calcowa math team
Enter your equation: a x² + b x + c = 0
x² − 3x + 2 = 0
For a quadratic, a can't be zero. Enter a non-zero a.
x = (3 ± √1) ÷ 2
What is the quadratic formula?
The quadratic formula is the one tool that solves every quadratic equation, no matter the numbers. A quadratic is any equation you can write as ax² + bx + c = 0, and the formula hands you the values of x that make it true. It works even when factoring is hard or impossible, which is why it's worth knowing by heart.
How do you use the quadratic formula?
Here's how to solve x² − 3x + 2 = 0:
- 1
Read off a, b, cHere a = 1, b = −3, and c = 2.
- 2
Find the discriminant(−3)² − 4 × 1 × 2 = 9 − 8 = 1.
- 3
Apply the formulax = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2.
- 4
Take both signsx = 2 and x = 1, the two roots.
The discriminant: how many roots?
Before you solve, the part under the square root, b² − 4ac, tells you what to expect. If it's positive, you get two different real roots and the parabola crosses the x-axis twice. If it's exactly zero, there's one repeated root and the parabola just touches the axis at its vertex. If it's negative, the roots are complex and the parabola never reaches the axis. The calculator labels which case you're in.
Roots, vertex, and the parabola
Every quadratic draws a parabola, a smooth U-shaped curve. The roots are where it crosses the x-axis, which is why they're also called x-intercepts. The vertex is the turning point, sitting at x = −b ÷ 2a, and the curve is a mirror image on each side of the vertical line through it, the axis of symmetry. The graph above plots your equation and marks the roots and the vertex, so the numbers and the picture line up.
Frequently asked questions
Does every quadratic have two solutions?
In a sense, yes. A quadratic always has two roots, but they're sometimes equal, which looks like one answer, or they're a pair of complex numbers when the discriminant is negative. The plus-or-minus in the formula is what produces the two.
The quadratic formula solves any equation of the form ax² + bx + c = 0. It says x = (−b ± √(b² − 4ac)) ÷ 2a. The plus-or-minus gives the two solutions, and you read a, b, and c straight off the equation.
Write your equation as ax² + bx + c = 0, read off a, b, and c, then plug them into x = (−b ± √(b² − 4ac)) ÷ 2a. Work out the part under the square root first, take its root, then do the plus and the minus separately to get both answers. The calculator shows each step.
The discriminant is the b² − 4ac part under the square root, and it tells you what kind of roots you'll get. If it's positive there are two real roots, if it's zero there's one repeated root, and if it's negative the two roots are complex numbers.
The roots are the x-values that make the equation equal zero, also called solutions or x-intercepts because that's where the parabola crosses the x-axis. A quadratic has two roots, though they're sometimes equal, or a pair of complex numbers when the parabola never touches the axis.
The vertex is the turning point, the lowest point if the parabola opens up or the highest if it opens down. Its x-value is −b ÷ 2a, and you put that back into the equation to get the y-value. The calculator marks it on the graph.
Yes. When the discriminant is negative, the square root of a negative number gives an imaginary part, so the two roots are complex, written as a real part plus or minus an imaginary part. That means the parabola sits entirely above or below the x-axis.
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