Quadratic Formula Calculator

Solve any quadratic equation in the form ax² + bx + c = 0. Enter the coefficients a, b and c to get both roots — real or complex — with step-by-step working, the discriminant and the vertex of the parabola.

Solutions
x₁ = 2, x₂ = 1
Two real roots
Equation
x² − 3x + 2 = 0
Discriminant (b² − 4ac)
1
Number of roots
Two real roots
Factored form
(x − 2)(x − 1)
Vertex (h, k)
(1.5, -0.25)
Axis of symmetry
x = 1.5
Parabola opens
Upward (minimum at the vertex)
y-intercept
(0, 2)
Sum of roots (−b/a)
3
Product of roots (c/a)
2
How it was calculated
  1. Formula: x = (−b ± √(b² − 4ac)) ÷ 2a
  2. Discriminant: b² − 4ac = (-3)² − 4 × 1 × 2 = 1
  3. √1 = 1, so x = (3 ± 1) ÷ 2
  4. x₁ = 2 and x₂ = 1

How to use the quadratic formula calculator

  1. Write your equation in standard form, ax² + bx + c = 0. Move every term to one side if needed.
  2. Enter the coefficients a, b and c. Use negative numbers for subtracted terms, and 0 for a missing term.
  3. See the roots, the step-by-step solution and key facts about the parabola.

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

The expression under the square root, b² − 4ac, is the discriminant. It tells you what kind of roots to expect:

  • Positive: two different real roots — the parabola crosses the x-axis twice.
  • Zero: one repeated real root — the vertex touches the x-axis.
  • Negative: two complex conjugate roots — the parabola never crosses the x-axis.

Worked example

Solve x² − 3x + 2 = 0. Here a = 1, b = −3, c = 2.

  1. Discriminant: (−3)² − 4 × 1 × 2 = 9 − 8 = 1.
  2. x = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2.
  3. x₁ = 4 ÷ 2 = 2 and x₂ = 2 ÷ 2 = 1. In factored form: (x − 2)(x − 1) = 0.

For x² + 2x + 5 = 0, the discriminant is 4 − 20 = −16, so the roots are complex: x = (−2 ± 4i) ÷ 2 = −1 ± 2i.

Vertex of a parabola

The graph of y = ax² + bx + c is a parabola with its vertex at x = −b ÷ 2a. Substitute that x back into the equation to get the y-coordinate. If a is positive the parabola opens upward and the vertex is the minimum; if a is negative it opens downward and the vertex is the maximum.

Other ways to solve quadratics

  • Factoring — fastest when the roots are whole numbers: x² − 5x + 6 = (x − 2)(x − 3).
  • Completing the square — rewrites the equation as a(x − h)² + k = 0.
  • Graphing — the roots are where the parabola crosses the x-axis.

The quadratic formula always works, which makes it the most reliable method.

Frequently asked questions

What is the quadratic formula?

For an equation ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. The ± gives the two roots.

What does the discriminant tell you?

The discriminant b² − 4ac shows how many real solutions there are: two if it is positive, one if it is zero and none (two complex roots) if it is negative.

Can a quadratic equation have no solution?

It always has two solutions when complex numbers are allowed, but it has no real solutions when the discriminant is negative — the parabola does not cross the x-axis.

What if a = 0?

Then the equation is not quadratic but linear: bx + c = 0, with the single solution x = −c ÷ b. The calculator handles this case automatically.

How do I find the vertex of a quadratic?

The vertex is at x = −b ÷ 2a. Plug that value into the equation to find y. For y = x² − 3x + 2, the vertex is at (1.5, −0.25).

Results are rounded for display. For academic or professional work, double-check critical calculations and follow the rounding rules your teacher, school or organization requires.

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