Root of Equations Calculator

Find the roots of linear, quadratic, and cubic equations with our Root of Equations Calculator. This tool helps you solve for x in equations of the form ax + b = 0, ax² + bx + c = 0, and ax³ + bx² + cx + d = 0.

What is a Root of an Equation?

A root of an equation is a solution that satisfies the equation, making it true when substituted for the variable. For example, in the equation x + 3 = 5, the root is x = 2 because 2 + 3 equals 5.

Equations can have one, two, or multiple roots depending on their type. Some equations may have complex roots involving imaginary numbers.

Key Concepts

Linear equations have one root
Quadratic equations can have two roots
Cubic equations can have three roots
Roots can be real or complex numbers

How to Use This Calculator

Using our Root of Equations Calculator is simple:

Select the type of equation you want to solve (linear, quadratic, or cubic)
Enter the coefficients for the equation
Click "Calculate" to find the roots
View the results and chart visualization

Tip

For quadratic equations, you can also view the graph of the parabola to visualize the roots.

Types of Equations

Linear Equations

Linear equations have the form ax + b = 0. They have exactly one root, which can be found using the formula:

Linear Equation Solution

x = -b / a

Quadratic Equations

Quadratic equations have the form ax² + bx + c = 0. They can have two real roots, one double root, or two complex roots. The roots can be found using the quadratic formula:

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

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. Solving cubic equations requires more advanced methods.

Example Calculations

Linear Equation Example

Find the root of 3x + 5 = 0:

Solution

x = -5 / 3 ≈ -1.6667

Quadratic Equation Example

Find the roots of x² - 5x + 6 = 0:

Solution

x = [5 ± √(25 - 24)] / 2

x₁ = (5 + 1)/2 = 3

x₂ = (5 - 1)/2 = 2

Cubic Equation Example

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

Solution

This equation has roots at x = 1, x = 2, and x = 3.

FAQ

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 values that satisfy the equation.

Can quadratic equations have complex roots?

Yes, quadratic equations can have complex roots when the discriminant (b² - 4ac) is negative. These roots involve imaginary 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, depending on the coefficients.

What is the discriminant in quadratic equations?

The discriminant is the part of the quadratic formula under the square root (b² - 4ac). It determines the nature of the roots.