Quadratic Formula Calculator

1. Write the equation in standard form. Rearrange to ax² + bx + c = 0. Every term must be on one side. For 2x² = 5x - 3, rearrange to 2x² - 5x + 3 = 0, so a = 2, b = -5, c = 3.
2. Identify a, b, and c. The coefficient of x² is a, the coefficient of x is b, and the constant term is c. Watch the signs: in x² - 5x + 6 = 0, b = -5 (negative) and c = +6.
3. Enter the coefficients. The calculator fills in the discriminant, roots, vertex, and step-by-step solution. If the discriminant is negative, the roots are complex (no real solutions).

The discriminant (b² - 4ac) tells you the solution type before you even calculate: positive gives two real roots, zero gives one repeated root, negative gives complex roots with no x-intercepts on a graph.

Standard form:     ax² + bx + c = 0

Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)

Discriminant:      D = b² - 4ac
D > 0: two distinct real roots
D = 0: one repeated real root (x = -b/2a)
D < 0: two complex conjugate roots

Vertex of parabola: (-b/2a, c - b²/4a)

Worked example: solve 2x² - 7x + 3 = 0