Roots Calculator X 3 X 2

How to Use This Calculator

Using the roots calculator is simple:

1. Enter the coefficients for the equation x³ + 2x = 0. In this case, the coefficients are already set to 1 for x³ and 2 for x.
2. Click the "Calculate" button to find all roots of the equation.
3. Review the results, which will show all real and complex roots.
4. Use the chart to visualize the roots on the complex plane.

The calculator will display all roots, including any repeated roots or complex solutions. The results are presented in both decimal and exact form where possible.

Formula Explained

The roots of the equation x³ + 2x = 0 can be found using the cubic formula. The general form of a cubic equation is:

For our specific equation x³ + 2x = 0, the coefficients are:

• a = 1 (coefficient of x³)
• b = 0 (coefficient of x²)
• c = 2 (coefficient of x)
• d = 0 (constant term)

The cubic formula for this equation is complex, but it can be simplified for our specific case. The roots are:

These are the three real roots of the equation x³ + 2x = 0.

Worked Example

Let's solve the equation x³ + 2x = 0 step by step.

1. Factor the equation: x(x² + 2) = 0
2. Set each factor equal to zero:
   • x = 0
   • x² + 2 = 0 → x² = -2 → x = ±√(-2) = ±i√2

The roots are x = 0, x = √2, and x = -√2. The calculator will display these roots in both decimal and exact form.

Interpreting the Results

The roots calculator provides several types of results:

• Real roots: These are the values of x that satisfy the equation in the real number system.
• Complex roots: These are the values of x that satisfy the equation in the complex number system.
• Multiplicity: The calculator indicates if any roots are repeated (have multiplicity greater than one).

For the equation x³ + 2x = 0, there are three real roots: x = 0, x = √2, and x = -√2. There are no complex roots in this case.