Roots of Second Degree Polynomial Calculator

What is a Second Degree Polynomial?

A second degree polynomial, also known as a quadratic equation, is a polynomial equation of the form:

where a, b, and c are constants, and a ≠ 0. The roots of the polynomial are the values of x that satisfy the equation. These roots can be real or complex numbers, depending on the discriminant.

Second degree polynomials are fundamental in algebra and have applications in physics, engineering, and economics.

Formula for Roots

The roots of a quadratic equation can be found using the quadratic formula:

The discriminant (D) is the part under the square root:

The nature of the roots depends on the discriminant:

• If D > 0: Two distinct real roots
• If D = 0: One real root (repeated)
• If D < 0: Two complex conjugate roots

How to Use the Calculator

To use the calculator:

1. Enter the coefficients a, b, and c of your quadratic equation
2. Click "Calculate Roots"
3. View the results including the roots and discriminant
4. Interpret the results based on the discriminant value

Worked Examples

Example 1: Two Distinct Real Roots

Find the roots of x² - 5x + 6 = 0.

Using the quadratic formula:

Roots: x = 3 and x = 2

Example 2: One Real Root

Find the roots of x² - 6x + 9 = 0.

Using the quadratic formula:

Root: x = 3 (repeated)

Example 3: Complex Roots

Find the roots of x² + 2x + 5 = 0.

Using the quadratic formula:

Roots: x = -1 + 2i and x = -1 - 2i

Interpreting Results

The calculator provides several key pieces of information:

• Roots: The solutions to the quadratic equation
• Discriminant: Indicates the nature of the roots
• Graph: Visual representation of the quadratic function

Understanding the discriminant helps determine the behavior of the quadratic function:

• Positive discriminant: The parabola intersects the x-axis at two points
• Zero discriminant: The parabola touches the x-axis at one point
• Negative discriminant: The parabola does not intersect the x-axis (no real roots)