Root of Equation Calculator

Find the roots of linear, quadratic, and cubic equations with our Root of Equation Calculator. This tool helps you solve equations of the form axⁿ + bx^n-1 + ... + k = 0.

What is a Root of an Equation?

A root of an equation is a solution to the equation. For a polynomial equation, roots are the values of x that make the equation equal to zero. For example, in the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3.

General Form: axⁿ + bx^n-1 + ... + k = 0

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

Types of Roots

Real Roots: Solutions that are real numbers.
Complex Roots: Solutions that include imaginary numbers (for higher-degree equations).
Repeated Roots: Roots that appear more than once (multiplicity).

Importance of Roots

Roots are essential in solving problems in physics, engineering, economics, and many other fields. They help determine the points where a function crosses the x-axis.

How to Use the Calculator

Select the type of equation (linear, quadratic, or cubic).
Enter the coefficients for the equation.
Click "Calculate" to find the roots.
Review the results and chart (if available).

For complex roots, the calculator will display them in the form a + bi, where i is the imaginary unit.

Formula Used

The calculator uses different methods depending on the equation type:

Linear Equation (ax + b = 0)

Root = -b / a

Quadratic Equation (ax² + bx + c = 0)

Roots = [-b ± √(b² - 4ac)] / (2a)

Cubic Equation (ax³ + bx² + cx + d = 0)

Roots are calculated using the cubic formula, which involves complex calculations.

Worked Examples

Example 1: Linear Equation

Find the root of 3x + 9 = 0.

Root = -9 / 3 = -3

Example 2: Quadratic Equation

Find the roots of x² - 5x + 6 = 0.

Roots = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2

Roots: x = 3 and x = 2

Example 3: Cubic Equation

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

Roots: x = 1, x = 2, x = 3

FAQ

What is the difference between a root and a solution?: A root is a solution to an equation. For polynomial equations, roots are the values of x that make the equation equal to zero.
Can all equations have real roots?: No, some equations have complex roots, especially higher-degree equations, which may require imaginary numbers.
How do I know if an equation has repeated roots?: Repeated roots occur when the discriminant of a quadratic equation is zero, or when a polynomial has a factor raised to a power greater than one.
What if the calculator shows complex roots?: Complex roots are valid solutions and are expressed in the form a + bi, where i is the imaginary unit.
Can this calculator solve equations with more than three terms?: Currently, the calculator supports linear, quadratic, and cubic equations. For more complex equations, advanced mathematical software may be needed.