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Quadratic Root Calculator with Steps

Reviewed by Calculator Editorial Team

This quadratic root calculator solves quadratic equations of the form ax² + bx + c = 0 and provides step-by-step solutions. Whether you're a student studying algebra or a professional needing quick solutions, this tool helps you find the roots of any quadratic equation.

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x with at least one x² term. The general form is:

ax² + bx + c = 0

Where:

  • a, b, and c are constants
  • a ≠ 0 (if a = 0, the equation is linear, not quadratic)
  • x is the variable

Quadratic equations can represent many real-world situations, such as projectile motion, area problems, and financial calculations.

How to Solve Quadratic Equations

There are several methods to solve quadratic equations:

  1. Factoring
  2. Completing the square
  3. Quadratic formula
  4. Graphical method

The quadratic formula is the most versatile method and works for all quadratic equations. This calculator uses the quadratic formula to find the roots.

The Quadratic Formula

The quadratic formula provides the roots of any quadratic equation ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • √(b² - 4ac) is the discriminant
  • If the discriminant is positive, there are two real roots
  • If the discriminant is zero, there is one real root
  • If the discriminant is negative, there are two complex roots

The calculator uses this formula to compute the roots and provides step-by-step explanations.

Example Calculation

Let's solve the quadratic equation x² - 5x + 6 = 0 using the quadratic formula.

  1. Identify the coefficients: a = 1, b = -5, c = 6
  2. Calculate the discriminant: b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
  3. Apply the quadratic formula:
    x = [5 ± √1] / 2
  4. Find the two roots:
    • x₁ = (5 + 1)/2 = 3
    • x₂ = (5 - 1)/2 = 2

The roots of the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

FAQ

What is the difference between a quadratic equation and a linear equation?
A quadratic equation has an x² term and can have two solutions, while a linear equation has only an x term and has exactly one solution.
How do I know if a quadratic equation has real roots?
A quadratic equation has real roots if the discriminant (b² - 4ac) is positive. If the discriminant is zero, there's exactly one real root. If it's negative, the roots are complex.
Can the quadratic formula be used for all quadratic equations?
Yes, the quadratic formula can solve any quadratic equation as long as a ≠ 0. It's the most reliable method for finding roots.