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Quadratic Equation Calculator

Struggling with a quadratic equation? Our calculator instantly finds the roots, calculates the discriminant, tells you whether the roots are real or complex, and breaks down the parabola's key properties — perfect for students and quick problem-solving.

Quadratic Equation Calculator

Solve equations of the form ax² + bx + c = 0, find roots, discriminant, and vertex instantly.

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How the Quadratic Equation Calculator Works

1

Enter Coefficients

Enter values for a, b, and c.

2

Calculate Discriminant

The calculator finds b² - 4ac.

3

Find Roots

Roots are calculated using the quadratic formula.

4

View Results

See roots, discriminant, and root type.

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

A quadratic equation is written as ax² + bx + c = 0. The quadratic formula calculates the roots of the equation using the coefficients a, b, and c.

Example Calculation

Input: a=1, b=-5, c=6

Output: Roots = 2 and 3

Common Uses

  • • Algebra
  • • Mathematics Education
  • • Engineering
  • • Physics
  • • Competitive Exams
  • • Graph Analysis
  • • Research
  • • Polynomial Equations

Questions about this calculator

A quadratic equation is a second-degree polynomial equation written as ax² + bx + c = 0.

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a.

The discriminant is b² - 4ac and determines the nature of the roots.

The equation has two distinct real roots.

The equation has one repeated real root.

The equation has two complex roots.

The x-coordinate of the vertex is given by x = -b / (2a). Substituting this x-value back into the quadratic function y = ax² + bx + c gives the corresponding y-coordinate.

Solve ax² + bx + c = 0

Enter the three coefficients a, b, and c, where a must not be zero. The calculator returns both roots, the discriminant, the root type, and the vertex of the related parabola y = ax² + bx + c.

Quadratic formula and discriminant

The roots are x = (-b ± √(b² - 4ac)) / (2a). The expression D = b² - 4ac is the discriminant. D > 0 gives two distinct real roots, D = 0 gives one repeated real root, and D < 0 gives a complex-conjugate pair.

Worked example

For x² - 5x + 6 = 0, a = 1, b = -5, and c = 6. The discriminant is 25 - 24 = 1. Substitution in the quadratic formula gives roots (5 + 1)/2 = 3 and (5 - 1)/2 = 2.

Vertex and common mistakes

The vertex x-coordinate is -b/(2a), and its y-coordinate is found by substituting that value into the polynomial. Keep the sign of each coefficient when entering it: in x² - 5x + 6, b is -5, not 5. If a is zero, the expression is linear rather than quadratic and this calculator will not solve it.

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Written & maintained by Aditya Singh

Aditya Singh is a software engineer and the founder of DevCalc, based in Uttar Pradesh, India. He builds and maintains every calculator on this site using documented formulas, worked examples, and clearly stated limitations, with careful attention to accuracy. Read more about the author →