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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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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

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.

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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