Quadratic Equation Solver

Solve ax² + bx + c = 0 with step-by-step discriminant, real, and complex roots

Calculation Result

Roots of Equation

x = 3, x = 2

Two real roots: x₁ = 3, x₂ = 2

Discriminant (D)

Vertex (h, k)

(2.5, -0.25)

Type

two-real-roots

Input coefficients a, b, and c for equation ax² + bx + c = 0.

Get root values, discriminant sign, and vertex coordinates (h, k).

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

Where D = b² - 4ac is the discriminant determining real, duplicate, or complex roots.

a = 1, b = -7, c = 12.

• Discriminant D = (-7)² - 4(1)(12) = 49 - 48 = 1

• x₁ = (7 + 1) / 2 = 4

• x₂ = (7 - 1) / 2 = 3

Two real roots: x = 4 and x = 3.

When D < 0, the equation has no real roots and instead produces two conjugate complex roots containing imaginary unit i.

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