The Ultimate Guide to Finding Roots

In mathematics, a "root" can have two common meanings. The first is the n-th root of a number, like a square root or cube root. The second, more advanced meaning is the root of an equation or function, which is the value that makes the equation true (equal to zero). This guide explains both concepts and shows how our versatile calculator can handle each type of problem with ease, providing step-by-step solutions for complex quadratic equations.

[Image of the quadratic formula]

How to Use the Root Calculator

Our tool has two distinct functions. Use the tabs to select the one you need.

1. N-th Root Calculator

This is used to find a root of a number, such as the square root (2nd root) of 9, which is 3.

  1. Enter the degree of the root (e.g., 2 for square root, 3 for cube root).
  2. Enter the number you want to find the root of.
  3. Click "Calculate".

2. Quadratic Equation Root Calculator

This is a polynomial root calculator specifically for second-degree equations in the form ax² + bx + c = 0. The "roots" are the values of 'x' that solve the equation.

  1. Enter the coefficients a, b, and c from your equation.
  2. Click "Find Roots".
  3. The calculator will display the roots and show a detailed, step-by-step application of the quadratic formula.

Finding the Roots of a Quadratic Equation

The roots of a quadratic equation are found using the famous quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

The part inside the square root, b² - 4ac, is called the "discriminant" (Δ). It tells us the nature of the roots:

  • If Δ > 0, there are two distinct real roots.
  • If Δ = 0, there is exactly one real root.
  • If Δ < 0, there are two complex (imaginary) roots.

Our calculator correctly handles all three cases and provides the full solution.

Example Calculation with Steps

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

  • a = 1, b = -5, c = 6
  • Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
  • x = [ -(-5) ± √1 ] / (2 * 1)
  • x = (5 ± 1) / 2
  • The two roots are x₁ = (5 + 1) / 2 = 3 and x₂ = (5 - 1) / 2 = 2.

Frequently Asked Questions (FAQ)

How does the root calculator for a quadratic equation work?

Select the 'Quadratic Equation' tab and enter the coefficients a, b, and c from your equation (ax² + bx + c = 0). The calculator uses the quadratic formula, x = [-b ± √(b²-4ac)] / 2a, to find the roots and shows you each step of the calculation.

Can this find polynomial roots?

This calculator is specifically designed to find the roots of second-degree polynomials, which are known as quadratic equations. It is the most common type of polynomial root problem encountered in algebra.

What is the formula for the n-th root?

The n-th root of a number 'y' is a number 'x' such that xⁿ = y. The calculator finds this by computing y^(1/n).

For more advanced calculations, see our Scientific Calculator. To find the slope of a line from an equation, use our Slope Calculator.