Write A Quartic Equation with The Given Roots Calculator

This calculator helps you construct a quartic equation (fourth-degree polynomial) from its given roots. Whether you're studying algebra, preparing for exams, or solving real-world problems, this tool provides a quick and accurate way to write quartic equations based on their roots.

How to Use This Calculator

Using our quartic equation calculator is simple:

Enter the four roots of your quartic equation in the input fields provided.
Click the "Calculate" button to generate the quartic equation.
Review the result, which will be displayed in both factored form and expanded form.
Use the optional chart to visualize the roots on the number line.

The calculator handles both real and complex roots, providing accurate results for all cases.

How It Works

A quartic equation is a polynomial equation of degree four. If you know the roots of the equation, you can write it in factored form as:

(x - r₁)(x - r₂)(x - r₃)(x - r₄) = 0

Where r₁, r₂, r₃, and r₄ are the roots of the equation. To convert this to standard form, you multiply out the factors:

x⁴ - (r₁ + r₂ + r₃ + r₄)x³ + (r₁r₂ + r₁r₃ + r₁r₄ + r₂r₃ + r₂r₄ + r₃r₄)x² - (r₁r₂r₃ + r₁r₂r₄ + r₁r₃r₄ + r₂r₃r₄)x + r₁r₂r₃r₄ = 0

Our calculator performs these calculations automatically, saving you time and reducing the chance of errors.

Note: For complex roots, the calculator will display them in the form a + bi, where a and b are real numbers, and i is the imaginary unit.

Worked Example

Let's say you have a quartic equation with roots at x = 1, x = -2, x = 3, and x = -4. Here's how you would write the equation:

Start with the factored form: (x - 1)(x + 2)(x - 3)(x + 4) = 0
Multiply the factors to get the expanded form:

x⁴ - (1 - 2 - 3 + 4)x³ + (1*(-2) + 1*(-3) + 1*4 + (-2)*(-3) + (-2)*4 + (-3)*4)x² - (1*(-2)*(-3) + 1*(-2)*4 + 1*(-3)*4 + (-2)*(-3)*4)x + 1*(-2)*(-3)*4 = 0
Simplify the coefficients to get:

x⁴ - (0)x³ + (-11)x² - (-28)x + 24 = 0
Which simplifies to: x⁴ - 11x² + 28x + 24 = 0

Our calculator would produce the same result for these roots.

Frequently Asked Questions

What is a quartic equation?: A quartic equation is a polynomial equation of degree four, typically written in the form ax⁴ + bx³ + cx² + dx + e = 0.
Can this calculator handle complex roots?: Yes, the calculator can handle both real and complex roots, displaying them in the standard a + bi format.
How accurate are the results?: The calculator uses precise mathematical operations to ensure accurate results for all valid inputs.
Can I use this calculator for educational purposes?: Absolutely! This calculator is designed to help students and professionals understand how to construct quartic equations from their roots.
Is there a limit to the size of the roots I can enter?: The calculator can handle roots of any size, as long as they are valid numbers or complex numbers in the form a + bi.