Cal11 calculator
Search calculator...

Identify The Roots of The Equation Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the roots of quadratic, cubic, and quartic equations. Roots are the solutions to an equation that make the equation true. They are also called solutions, zeros, or x-intercepts.

What are roots of an equation?

The roots of an equation are the values of the variable that satisfy the equation. For example, in the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3 because these values make the equation true.

Roots can be real or complex numbers. Real roots are points where the graph of the equation crosses the x-axis. Complex roots come in pairs and are not real numbers.

How to find roots of an equation

There are several methods to find the roots of an equation:

  1. Factoring: Express the equation as a product of factors and set each factor equal to zero.
  2. Quadratic Formula: For quadratic equations (ax² + bx + c = 0), use the formula x = [-b ± √(b² - 4ac)] / (2a).
  3. Cubic Formula: For cubic equations (ax³ + bx² + cx + d = 0), use Cardano's formula.
  4. Numerical Methods: Approximate roots using methods like the Newton-Raphson method.
  5. Graphical Methods: Plot the equation and identify where it crosses the x-axis.

Quadratic Formula

For an equation ax² + bx + c = 0, the roots are:

x = [-b ± √(b² - 4ac)] / (2a)

Types of equations and their roots

Quadratic Equations

Quadratic equations have the form ax² + bx + c = 0. They can have two real roots, one real root (a repeated root), or two complex roots.

Cubic Equations

Cubic equations have the form ax³ + bx² + cx + d = 0. They can have one real root and two complex roots, or three real roots (which may be repeated).

Quartic Equations

Quartic equations have the form ax⁴ + bx³ + cx² + dx + e = 0. They can have up to four real roots, or two real roots and two complex roots.

Example calculations

Quadratic Equation Example

Find the roots of x² - 5x + 6 = 0.

  1. Identify coefficients: a = 1, b = -5, c = 6.
  2. Calculate discriminant: D = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1.
  3. Apply quadratic formula: x = [5 ± √1] / 2.
  4. Roots: x = (5 + 1)/2 = 3 and x = (5 - 1)/2 = 2.

Cubic Equation Example

Find the roots of x³ - 6x² + 11x - 6 = 0.

  1. Factor the equation: (x - 1)(x - 2)(x - 3) = 0.
  2. Set each factor to zero: x = 1, x = 2, x = 3.

Frequently Asked Questions

What is the difference between a root and a solution?
In the context of equations, "root" and "solution" are often used interchangeably. Both refer to the values that satisfy the equation.
Can an equation have complex roots?
Yes, if the discriminant of a quadratic equation is negative, the roots will be complex numbers.
How many roots can a cubic equation have?
A cubic equation can have one real root and two complex roots, or three real roots (which may be repeated).
What is the difference between a repeated root and a double root?
A repeated root is a root that occurs more than once. A double root is a specific case of a repeated root where the root occurs exactly twice.
How can I find the roots of an equation if it's not factorable?
You can use numerical methods like the Newton-Raphson method or graphical methods to approximate the roots.