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Quadratic Formula Calculator

Solve ax² + bx + c = 0 and see every step, including the discriminant, the vertex and complex roots.

Results

ItemValue

Steps

StepWorking
Answers are rounded to 6 decimal places.

Quick Answer

The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. For x² − 3x + 2 = 0, the discriminant is 1, so x = 2 or x = 1. A positive discriminant gives two real roots, zero gives one, and a negative value gives two complex roots.

Last updated: 2 November 2026 ·How we calculate this: the quadratic formula

Table of Contents

How This Quadratic Formula Calculator Works

A quadratic equation has the form ax² + bx + c = 0. Enter a, b and c, and the calculator finds the roots, the discriminant and the vertex. It also shows each step, so you can follow the working.

Quick explanation: the discriminant, b² − 4ac, tells you what type of roots to expect.

  • Positive: two different real roots.
  • Zero: one repeated real root.
  • Negative: two complex roots.

How to Use the Quadratic Formula Calculator

  1. Rearrange your equation so it reads ax² + bx + c = 0.
  2. Enter a, b and c, including any minus signs. A missing term has a value of 0.
  3. Press Calculate.
  4. Read the roots, the discriminant and the vertex in the results.
  5. Follow the steps table to check your own working.

Core Formulas

  • Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a
  • Discriminant: D = b² − 4ac
  • Vertex x: −b ÷ 2a
  • Vertex y: c − b² ÷ 4a

Key Terms

  • Quadratic equation: an equation with x² as its highest power, written ax² + bx + c = 0.
  • Coefficients: the numbers a, b and c in front of x², x and the constant.
  • Root: a value of x that makes the equation true. It is also called a solution or zero.
  • Discriminant: the value b² − 4ac, which shows how many real roots there are.
  • Parabola: the U-shaped curve you get when you graph y = ax² + bx + c.
  • Vertex: the turning point of the parabola, at x = −b ÷ 2a.
  • Complex root: a root with an imaginary part, written a + bi, where i = √−1.

Worked Example

Solve x² + 2x + 5 = 0, so a = 1, b = 2 and c = 5.

  • Discriminant = 2² − 4(1)(5) = −16
  • √−16 = 4i
  • x = (−2 ± 4i) ÷ 2
  • x = −1 + 2i or x = −1 − 2i

Real-Life Examples

  • A thrown ball: a ball's height in feet is h = −16t² + 64t + 5. Setting h = 0 gives a = −16, b = 64, c = 5. The positive root, t = 4.08 seconds, is when it lands.
  • A garden plot: a rectangle is 4 m longer than it is wide and has an area of 60 m². That gives w² + 4w − 60 = 0, so the width is 6 m and the length 10 m.
  • Break-even point: if profit is −2x² + 40x − 150, solving −2x² + 40x − 150 = 0 gives x = 5 and x = 15. The business makes a profit between those two levels.

Quadratic Formula vs Factoring vs Completing the Square

  • Factoring: fastest when the equation splits into whole-number brackets, such as x² − 3x + 2 = (x − 1)(x − 2).
  • Completing the square: useful for finding the vertex form and for proving the formula.
  • Quadratic formula: always works, including for decimals and complex roots.
  • Graphing: shows the roots as the points where the curve crosses the x-axis.

What the Discriminant Tells You

Discriminant (b² − 4ac)RootsGraph
PositiveTwo different real rootsCrosses the x-axis twice
ZeroOne repeated real rootTouches the x-axis at the vertex
NegativeTwo complex rootsDoes not touch the x-axis
A perfect squareTwo rational rootsThe equation can be factored

Common Mistakes

  • Equation not equal to zero: move all terms to one side first.
  • Negative signs: if b is negative, −b becomes positive.
  • Squaring a negative: (−3)² is 9, not −9.
  • Dividing only part: divide the whole top by 2a.

Frequently Asked Questions

It is x = (−b ± √(b² − 4ac)) ÷ 2a, used to solve any equation of the form ax² + bx + c = 0 where a is not zero.

It shows the type of roots: positive gives two real roots, zero gives one, and negative gives two complex roots.

If a is 0, the x² term disappears, so the equation is linear and not quadratic.

Yes. If the discriminant is negative, it shows the two roots in a + bi form.

It is the highest or lowest point of the graph of y = ax² + bx + c. Its x value is −b ÷ 2a.

If the discriminant is a perfect square, such as 1, 4 or 9, the equation factors with rational numbers.

The roots add up to −b ÷ a and multiply to c ÷ a. This is a quick way to check your answer.

It is y = a(x − h)² + k, where (h, k) is the vertex. This calculator gives h and k for you.

References & Sources

SourceUsed ForLink
Wikipedia, Quadratic formula The formula and its derivation. https://en.wikipedia.org/wiki/Quadratic_formula
Wikipedia, Discriminant How the discriminant describes the roots. https://en.wikipedia.org/wiki/Discriminant