Quadratic Equation Solver

No. 4 — ax² + bx + c = 0
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a·x² + b·x + c = 0
x² + x + = 0
x = 2 and x = 1 REAL ROOTS
Discriminant Δ = 1 · Vertex (1.5, −0.25)

How the math works

The quadratic formula

For any quadratic ax² + bx + c = 0, the two solutions come from:

x = (−b ± √(b² − 4ac)) / 2a

The discriminant

Let Δ = b² − 4ac. Its sign decides the nature of the roots:

Δ > 0 → two real roots · Δ = 0 → one repeated root · Δ < 0 → two complex roots

The vertex

The turning point of the parabola sits halfway between the two roots:

vertex x = −b / 2a

Frequently asked

What if a is zero? Then it's not a quadratic — it's a linear equation, and the quadratic formula breaks down. Enter a non-zero a.

What do complex roots look like? When Δ is negative, the roots involve the imaginary unit i, such as x = 1 ± 2i.