Quadratic Formula Calculator

Enter coefficients a, b, and c for any quadratic equation ax² + bx + c = 0 to find real or complex roots, discriminant (Δ), parabola vertex, and y-intercept.

How to use this calculator

1. Enter coefficient a (the multiplier of x²).
2. Enter coefficient b (the multiplier of x).
3. Enter constant c.
4. Read your primary root (x₁), secondary root (x₂), discriminant Δ (b² - 4ac), and parabola vertex (h, k).

Formula & method

Quadratic Formula & Discriminant Equations

x = [ −b ± √(b² − 4ac) ] / (2a) · Discriminant Δ = b² − 4ac · Vertex (h, k) = ( −b/(2a) , c − b²/(4a) )

a ≠ 0. If Δ < 0, roots are expressed in standard complex form a + bi.

Example

x² − 5x + 6 = 0 (a=1, b=-5, c=6) → Δ = 25 − 24 = 1 > 0 → x₁ = (5+1)/2 = 3, x₂ = (5-1)/2 = 2. Vertex: (2.5, -0.25).

Key insights

• The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 yields 2 distinct real roots, Δ = 0 yields 1 repeated real root, and Δ < 0 yields 2 complex conjugate roots.
• The vertex of the quadratic parabola ax² + bx + c occurs at x = −b / (2a); if a > 0 the parabola opens upwards, if a < 0 it opens downwards.
• Quadratic equations model physical phenomena including projectile motion, acceleration, satellite dish geometry, and profit optimization curves.
• By Vieta's formulas, the sum of the roots x₁ + x₂ = −b/a and the product x₁ × x₂ = c/a.

How to interpret your result

Roots x₁ and x₂

The x-intercepts where the quadratic graph intersects the horizontal axis (ax² + bx + c = 0).

Discriminant (Δ)

Indicates whether the parabola crosses the x-axis twice (Δ > 0), touches it at a single point (Δ = 0), or hovers above/below without touching (Δ < 0).

Parabola Vertex (h, k)

The minimum point (if a > 0) or maximum peak (if a < 0) of the parabolic function.

Frequently asked questions

What happens if a = 0 in a quadratic equation?

If a = 0, the equation reduces to a linear equation bx + c = 0 with a single solution x = -c/b.

What are complex roots?

Complex roots occur when the discriminant Δ < 0. They involve the imaginary unit i (where i = √-1) and take the form x = α ± βi.

How do I find the peak (maximum) of a parabola?

Calculate x = -b/(2a), then substitute x back into ax² + bx + c to find the maximum y value.

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