Quadratic Formula Calculator

Q: How do I find the vertex of a parabola?

For y = ax² + bx + c, the vertex sits at h = −b ÷ 2a with k = c − b² ÷ 4a. When a > 0 the parabola opens upward and the vertex is the minimum; when a < 0 it opens downward and the vertex is the maximum.

Q: What if a is zero?

Then the equation is not quadratic — it is the linear equation bx + c = 0 with the single root x = −c ÷ b. The quadratic formula requires a ≠ 0 because it divides by 2a.

Solve ax² + bx + c = 0 live as you type: roots, discriminant, vertex, and the worked solution — including complex roots when the discriminant is negative.

x² + x + = 0

Root x₁ -

Root x₂ -

Discriminant b² − 4ac -

Vertex -

Step-by-step solution

Enter coefficients above to see the working.

Formulas used

Quantity	Formula
Roots	x = (−b ± √(b² − 4ac)) ÷ 2a
Discriminant	D = b² − 4ac
Vertex	(h, k) = (−b ÷ 2a, c − b² ÷ 4a)
Factored form	ax² + bx + c = a(x − x₁)(x − x₂)

Frequently asked questions

What is the quadratic formula?

The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. It always works, even when the equation cannot be factored easily.

What does the discriminant tell me?

The discriminant D = b² − 4ac predicts the roots before you solve. D > 0 gives two distinct real roots, D = 0 gives one repeated real root, and D < 0 gives two complex conjugate roots.

How do I find the vertex of a parabola?

For y = ax² + bx + c, the vertex sits at h = −b ÷ 2a with k = c − b² ÷ 4a. When a > 0 the parabola opens upward and the vertex is the minimum; when a < 0 it opens downward and the vertex is the maximum.

What if a is zero?

Then the equation is not quadratic — it is the linear equation bx + c = 0 with the single root x = −c ÷ b. The quadratic formula requires a ≠ 0 because it divides by 2a.