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Quadratic Formula Calculator
Solve ax² + bx + c = 0 with Step-by-Step Solutions

Professional quadratic equation solver using the quadratic formula x = (-b ± sqrt(b²-4ac)) / 2a. Find real and complex roots, analyze the discriminant, and understand parabola properties.

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Detailed explanations

Quadratic Formula Calculator

Solve quadratic equations in the form ax² + bx + c = 0

Current Equation

x² - 5x + 6 = 0

Coefficient of x²

Coefficient of x

Constant term

Example Equations

Solution

Roots and properties of the quadratic equation

Enter valid coefficients to solve the quadratic equation

Understanding Quadratic Equations

Master the quadratic formula and understand what your solutions mean.

The Quadratic Formula
x = (-b ± sqrt(b² - 4ac)) / 2a. This universal formula solves any quadratic equation ax² + bx + c = 0 where a ≠ 0.
The Discriminant
Delta = b² - 4ac determines root types: positive = 2 real roots, zero = 1 repeated root, negative = 2 complex roots.
Parabola Vertex
The vertex is at x = -b/2a. This is the maximum (if a < 0) or minimum (if a > 0) point of the parabola.
Factoring Method
If roots are r and s, then ax² + bx + c = a(x - r)(x - s). Factoring is faster when roots are rational numbers.
Complex Roots
When discriminant < 0, roots are complex conjugates: a + bi and a - bi. The parabola doesn't cross the x-axis.
Real-World Applications
Quadratics model projectile motion, optimization problems, area calculations, and many physics applications.

Understanding the Discriminant

Delta > 0: Two Real Roots

  • Two distinct x-intercepts
  • Parabola crosses x-axis twice
  • Example: x² - 5x + 6 = 0
  • Roots: x = 2 and x = 3
  • Can be factored easily

Delta = 0: One Repeated Root

  • Parabola touches x-axis
  • Vertex is on the x-axis
  • Example: x² - 4x + 4 = 0
  • Root: x = 2 (multiplicity 2)
  • Perfect square trinomial

Delta < 0: Complex Roots

  • No real x-intercepts
  • Parabola above/below x-axis
  • Example: x² + 2x + 5 = 0
  • Roots: -1 + 2i and -1 - 2i
  • Complex conjugate pairs

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