Roots of A Quartic Equation Calculator

How to Use This Calculator

To find the roots of a quartic equation:

1. Enter the coefficients a, b, c, and d for the equation x⁴ + a x³ + b x² + c x + d = 0
2. Click "Calculate Roots"
3. View the results showing all real and complex roots
4. Use the chart to visualize the roots on the complex plane

The calculator will display all four roots (real or complex) of your quartic equation. For complex roots, it shows both the real and imaginary parts.

Formula Used

The calculator uses numerical methods to solve quartic equations when analytical solutions are not possible. The general approach involves:

1. Reducing the quartic to a depressed quartic (x⁴ + p x² + q x + r = 0)
2. Using Ferrari's method to find the roots
3. Solving the resulting cubic equation
4. Back-substituting to find all four roots

The exact numerical solution depends on the specific coefficients entered, but the calculator handles all cases including multiple real roots and complex conjugate pairs.

Worked Example

Let's solve x⁴ - 5x² + 4 = 0:

1. Set a = 0, b = -5, c = 0, d = 4
2. The roots are x = ±1 and x = ±2
3. The calculator will display these four real roots

This shows how the calculator can find all roots of a quartic equation, whether they are real or complex.

Interpreting Results

When you get results from the calculator:

• Real roots are straightforward solutions to the equation
• Complex roots come in conjugate pairs (a + bi and a - bi)
• Multiple roots indicate the equation has repeated solutions
• The chart helps visualize the distribution of roots on the complex plane

For practical applications, you may need to consider only real roots or the magnitude of complex roots depending on your specific problem.