Quadratic Formula Calculator - Calculator See


Quadratic Formula Calculator

Solve quadratic equations instantly with step-by-step factoring, real or complex roots, discriminant analysis, and parabola vertex coordinates.

Calculator Inputs

Quadratic coefficient (a ≠ 0).
Linear coefficient.
Constant term.

Calculated Results

Real Roots (x₁, x₂)
x₁ = 2.00, x₂ = -3.00
Discriminant (b² - 4ac) 25.00 (Two Real Roots)
Parabola Vertex (h, k) (-0.50, -6.25)
Y-Intercept (0, -6.00)
Values calculate live as you adjust inputs.

About Quadratic Formula Calculator

The quadratic formula provides the analytic solutions to any second-degree polynomial equation ax² + bx + c = 0, determining where the parabola crosses the x-axis.

What is Quadratic Formula Calculator?

The quadratic formula provides the analytic solutions to any second-degree polynomial equation ax² + bx + c = 0, determining where the parabola crosses the x-axis.

How to Use This Calculator

  1. Enter coefficients a, b, and constant c.
  2. The tool evaluates the discriminant (b² - 4ac).
  3. Examine roots (real or imaginary) and graph vertex coordinates.

The Mathematical Formula & Variables

x = (-b ± Sqrt(b^2 - 4*a*c)) / (2*a)

Variables Definition

Symbol / Variable Name Description
Δ Discriminant b² - 4ac determines root nature (>0 real distinct, =0 duplicate, <0 complex).

Step-by-Step Worked Example

Scenario: Solve x² + x - 6 = 0 (a=1, b=1, c=-6).

  1. Discriminant = 1² - 4(1)(-6) = 1 + 24 = 25.
  2. √25 = 5.
  3. x₁ = (-1 + 5) / 2 = 4 / 2 = 2.
  4. xâ‚‚ = (-1 - 5) / 2 = -6 / 2 = -3.

Result: x = 2 or x = -3

Frequently Asked Questions

What happens when the discriminant is negative?

When b² - 4ac < 0, the equation has no real solutions. Instead, it yields two complex conjugate roots involving the imaginary unit i (where i = √-1).