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Algebraic Suite

Quadratic Formula

Resolve second-degree polynomials of the form ax² + bx + c = 0. Our engine calculates the discriminant to determine root types, handles imaginary solutions, and provides a full geometric analysis of the resulting parabola.

Enter coefficients for the standard form ax² + bx + c = 0. Note that 'a' cannot be zero.

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Parabola Analysis

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The standard form is ax² + bx + c = 0, where x represents an unknown, and a, b, and c are constants.

The Quadratic Formula

x = [-b ± √(b² - 4ac)] / 2a

This formula works for all quadratic equations, regardless of whether the roots are real or complex numbers.

The Discriminant (Δ)

The term inside the square root, b² - 4ac, is known as the discriminant. It tells us the nature of the roots:

  • Δ>0
    Two Distinct Real RootsThe parabola crosses the x-axis at two different points.
  • Δ=0
    One Repeated Real RootThe parabola touches the x-axis at exactly one point (the vertex).
  • Δ<0
    Two Complex RootsThe parabola does not cross the x-axis. Solutions involve the imaginary unit (i).

Alternative Methods

Factoring

Finding two numbers that multiply to ac and add to b. Ideal for "clean" integer solutions.

Completing the Square

Transforming the equation into (x-h)² = k. This is the derivation source of the quadratic formula.