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Quadratic Formula Calculator

Solve ax² + bx + c = 0 using the quadratic formula

a =

b =

c =

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

The quadratic formula solves any equation in the form ax²+bx+c=0 where a is not zero. The expression b²-4ac is called the discriminant, and it controls the type of roots before the final division is done. A positive discriminant gives two real solutions, a zero discriminant gives one repeated real solution, and a negative discriminant gives two complex solutions. This makes the formula a reliable method even when factoring is difficult.

⚠Note: a must not be 0. If a=0, the expression is linear rather than quadratic.

What This Calculator Shows

Discriminant

Computes Δ=b²-4ac and explains the root type.

Real or Complex Roots

Displays exact root structure and decimal values when appropriate.

Vertex

Finds the x-coordinate -b/(2a), useful for graphing parabolas.

Vieta Check

Verifies that root sum equals -b/a and product equals c/a.

💡 Example: x²-5x+6=0 has Δ=1, so x=(5±1)/2. The roots are x=3 and x=2.

Applications

ParabolasProjectile MotionOptimizationAlgebra Homework

Frequently Asked Questions

What is a quadratic formula calculator?▼

A quadratic formula calculator solves equations in the form ax²+bx+c=0 using x=(-b±√(b²-4ac))/(2a).

What is the quadratic formula?▼

For ax²+bx+c=0, x=(-b±√(b²-4ac))/(2a).

What if Δ is negative?▼

There are no real roots. The two roots are complex conjugates.

What does the discriminant show?▼

The discriminant Δ=b²-4ac shows root type: positive gives two real roots, zero gives one repeated real root, and negative gives complex roots.

Can a be zero?▼

No. If a=0, the equation is linear, not quadratic. Use a linear equation solver instead.

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