Quadratic Formula Calculator

Step 1: Identify a, b, c from ax² + bx + c = 0.

Step 2: Enter a (cannot be zero), b, and c.

Step 3: Click Calculate for roots, discriminant, vertex, and step-by-step solution.

A quadratic equation has the form ax² + bx + c = 0 where a is not zero. Every quadratic has exactly two roots: two real, one repeated, or two complex.

The discriminant (b²-4ac) determines root type: positive = two real, zero = one repeated, negative = complex.

The graph is a parabola. Opens up if a>0, down if a<0. Vertex at x = -b/(2a).

x = (-b ± √(b²-4ac)) / (2a)

Discriminant:
- Positive: Two real roots
- Zero: One repeated root
- Negative: Two complex roots

Example: x²-5x+6=0 (a=1,b=-5,c=6)
D = 25-24 = 1
x = (5±1)/2, x1=3, x2=2

Example 1: x²-5x+6=0: D=1, x=3 and x=2

Example 2: x²-6x+9=0: D=0, x=3 (double root)

Example 3: x²+2x+5=0: D=-16, x=-1±2i (complex)

Try factoring first. x²-5x+6 = (x-2)(x-3) is faster.

a cannot be zero. That makes it linear, not quadratic.

Negative discriminant = complex roots. Always in conjugate pairs.

Vertex for optimization. x=-b/(2a) gives max/min.