Roots of Equation Calculator with Steps

Finding the roots of an equation is a fundamental problem in mathematics with applications in science, engineering, and finance. This calculator helps you find the roots of linear, quadratic, and cubic equations with detailed step-by-step solutions.

Introduction to Roots of Equations

The roots of an equation are the values of the variable that satisfy the equation. For a polynomial equation, these are the values of x that make the equation equal to zero. Different types of equations have different methods for finding their roots.

General Form of a Polynomial Equation

axⁿ + bx^n-1 + ... + k = 0

Where a, b, ..., k are coefficients and n is the degree of the polynomial.

For equations of degree 1 (linear), 2 (quadratic), and 3 (cubic), we have specific formulas to find the roots. Higher-degree equations typically require numerical methods.

How to Use the Calculator

Our calculator can solve linear, quadratic, and cubic equations. Follow these steps:

Select the type of equation you want to solve from the dropdown menu.
Enter the coefficients of the equation in the provided fields.
Click "Calculate" to see the roots and step-by-step solution.
Review the results and use the chart to visualize the equation and its roots.

Note

For quadratic equations, the calculator will show both real roots if they exist. For cubic equations, it will show all three roots, including complex ones if necessary.

Methods for Finding Roots

Different methods are used depending on the type of equation:

Linear Equations (Degree 1)

For equations of the form ax + b = 0, the root is simply:

Root of Linear Equation

x = -b / a

Quadratic Equations (Degree 2)

For equations of the form ax² + bx + c = 0, the roots can be found using the quadratic formula:

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) determines the nature of the roots:

If discriminant > 0: Two distinct real roots
If discriminant = 0: One real root (repeated)
If discriminant < 0: Two complex conjugate roots

Cubic Equations (Degree 3)

For equations of the form ax³ + bx² + cx + d = 0, the roots can be found using the cubic formula, which is more complex and involves solving a depressed cubic equation.

Worked Examples

Example 1: Linear Equation

Find the root of 3x + 5 = 0.

Using the linear equation formula:

x = -5 / 3 ≈ -1.6667

Example 2: Quadratic Equation

Find the roots of x² - 5x + 6 = 0.

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2

x = [5 ± 1] / 2

Roots: x = 3 and x = 2

Example 3: Cubic Equation

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

Using the cubic formula, we find the roots are x = 1, x = 2, and 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 of the variable that satisfy the equation.

Can this calculator solve equations with more than three variables?

This calculator is designed to solve polynomial equations with one variable. For systems of equations with multiple variables, you would need a different type of calculator.

What if the discriminant is negative for a quadratic equation?

If the discriminant is negative, the quadratic equation has two complex conjugate roots. The calculator will display these roots in the form a ± bi, where i is the imaginary unit.

How accurate are the results from this calculator?

The calculator uses standard mathematical formulas and provides accurate results for the types of equations it supports. For complex calculations, it may show results rounded to a reasonable number of decimal places.