Roots Calculator with Steps Mathpapa

How to Use This Calculator

Using our roots calculator is simple and straightforward. Follow these steps to find the roots of your equation:

1. Enter your polynomial equation in the input field. For example, you can enter "x² - 5x + 6" for a quadratic equation.
2. Select the degree of your equation from the dropdown menu.
3. Click the "Calculate" button to find the roots.
4. Review the detailed steps and the final roots displayed in the result section.

The calculator will provide step-by-step solutions, including the use of relevant formulas, to help you understand how the roots are calculated.

Formula Used

The roots of a polynomial equation can be found using various methods depending on the degree of the equation. For quadratic equations (degree 2), the quadratic formula is commonly used:

For higher-degree equations, methods such as factoring, synthetic division, or numerical methods like Newton's method may be used. The calculator automatically selects the appropriate method based on the equation's degree.

Worked Examples

Example 1: Quadratic Equation

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

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

The roots of the equation are x = 2 and x = 3.

Example 2: Cubic Equation

Let's solve the cubic equation x³ - 6x² + 11x - 6 = 0.

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

The roots of the equation are x = 1, x = 2, and x = 3.