Quadratic Equation From Complex Roots Calculator

This calculator helps you find the quadratic equation when you know its complex roots. A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants. When the roots are complex, they come in conjugate pairs, and the equation can be expressed in terms of these roots.

Introduction

Quadratic equations are fundamental in algebra and have wide applications in physics, engineering, and economics. When the roots of a quadratic equation are complex, they are complex conjugates of each other. This calculator allows you to reverse-engineer the quadratic equation from its complex roots.

The process involves using the roots to construct the equation in its factored form and then expanding it to the standard quadratic form. This is useful when you know the roots of a quadratic equation and need to find the original equation.

How to Use the Calculator

Using the calculator is straightforward. Follow these steps:

Enter the real part of the first complex root in the "Real part of first root" field.
Enter the imaginary part of the first complex root in the "Imaginary part of first root" field.
Enter the real part of the second complex root in the "Real part of second root" field.
Enter the imaginary part of the second complex root in the "Imaginary part of second root" field.
Click the "Calculate" button to generate the quadratic equation.

The calculator will display the quadratic equation in its standard form and provide a visualization of the roots.

Formula

If a quadratic equation has complex roots \( r_1 \) and \( r_2 \), it can be expressed in its factored form as:

Factored Form

\( a(x - r_1)(x - r_2) = 0 \)

To convert this to the standard quadratic form \( ax^2 + bx + c = 0 \), you need to expand the factored form. The coefficients \( a \), \( b \), and \( c \) can be determined based on the roots.

Standard Form

\( ax^2 - a(r_1 + r_2)x + a r_1 r_2 = 0 \)

In this formula, \( r_1 \) and \( r_2 \) are the complex roots, and \( a \) is a non-zero constant. The calculator uses this formula to derive the quadratic equation from the given roots.

Worked Example

Let's consider a quadratic equation with complex roots \( r_1 = 2 + 3i \) and \( r_2 = 2 - 3i \). We can use the calculator to find the quadratic equation.

Example Input

Real part of first root: 2

Imaginary part of first root: 3

Real part of second root: 2

Imaginary part of second root: -3

The calculator will generate the quadratic equation as follows:

Example Output

Quadratic Equation: x² - 4x + 13 = 0

This means the quadratic equation with roots \( 2 + 3i \) and \( 2 - 3i \) is \( x^2 - 4x + 13 = 0 \).

FAQ

What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.

What are complex roots?

Complex roots are roots of a quadratic equation that are complex numbers, meaning they have both a real and an imaginary part.

How do I find the quadratic equation from complex roots?

You can use the factored form of the quadratic equation and expand it to the standard form using the given complex roots.

Can the calculator handle non-conjugate complex roots?

No, the calculator assumes the roots are complex conjugates. If the roots are not conjugates, the resulting equation may not be quadratic.

Is the constant \( a \) included in the output?

Yes, the calculator assumes \( a = 1 \) for simplicity. If you need a different value for \( a \), you can multiply the equation by the desired constant.