Vertex Form Calculator

Vertex Form Calculator: solve vertex form problems step-by-step. Clear formula, worked example, and FAQs included.

A quadratic equation calculator solves equations of the form ax² + bx + c = 0, finding the values of x (roots) where the parabola crosses the x-axis. Quadratic equations appear throughout physics, engineering, economics, and geometry. The discriminant (b² − 4ac) determines the nature of the roots: positive = two real roots; zero = one repeated root; negative = two complex roots. Percentage Calculator and Ratio Calculator provide related mathematical tools.

Enter coefficients a, b, and c from your equation ax² + bx + c = 0.
The calculator computes the discriminant and applies the quadratic formula.
Read the roots x₁ and x₂ — these are the x-values where the quadratic equals zero.
If the discriminant is negative, the roots are complex (no real solutions).

Quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Discriminant Δ = b² − 4ac: Δ > 0 → two distinct real roots; Δ = 0 → one repeated root; Δ < 0 → two complex roots.

Example: x² − 5x + 6 = 0. a=1, b=−5, c=6. Δ = 25−24 = 1. x = (5 ± 1) ÷ 2. Roots: x₁ = 3, x₂ = 2.

Interpreting roots

Types of solutions

Two real roots mean the parabola crosses the x-axis twice. One repeated root means it touches but doesn't cross. Complex roots mean the parabola doesn't cross the x-axis — the equation has no real solutions. Vertex of the parabola: x = −b÷(2a), y = c − b²÷(4a).

Math tips and best practices

Always check a ≠ 0 — if a = 0 the equation is linear, not quadratic.
Factor first if possible — if the roots are integers, factoring is faster than the formula.
The sum of roots = −b/a; the product of roots = c/a — useful checks.

The quadratic formula was known to Babylonian mathematicians around 2000 BCE, making it one of the oldest mathematical results still in common use.
Quadratic equations model projectile motion: height = −½gt² + v₀t + h₀ where solving for t gives the time of flight.

Common mistakes to avoid

Sign errors in b — the formula uses −b, so a negative b becomes positive.
Forgetting to take the square root of the discriminant — compute √(b²−4ac), not b²−4ac.
Dividing by 2a not just 2 — the entire expression (−b ± √Δ) is divided by 2a.

Mathematical calculations are deterministic — results are exact for the inputs provided. Floating-point arithmetic in digital calculators may introduce small rounding errors for very large or very small coefficients. This calculator is for educational purposes.