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Second Order Polynomial Roots Calculator

Reviewed by Calculator Editorial Team

A second order polynomial, also known as a quadratic equation, is a polynomial equation of degree 2. The general form is ax² + bx + c = 0. This calculator finds the roots of such equations using the quadratic formula.

What is a Second Order Polynomial?

A second order polynomial is a quadratic equation that can be written in the standard form:

ax² + bx + c = 0

where a, b, and c are coefficients, and x is the variable.

Quadratic equations appear in many real-world problems, including physics, engineering, and economics. The roots of the equation represent the points where the quadratic function crosses the x-axis.

How to Calculate Roots

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

x = [-b ± √(b² - 4ac)] / (2a)

This formula provides two roots, which may be real and distinct, real and equal, or complex conjugates depending on the discriminant (b² - 4ac).

Steps to Calculate:

  1. Identify the coefficients a, b, and c from the quadratic equation.
  2. Calculate the discriminant: D = b² - 4ac.
  3. If D > 0, there are two distinct real roots.
  4. If D = 0, there is one real root (a repeated root).
  5. If D < 0, the roots are complex conjugates.
  6. Apply the quadratic formula to find the roots.

Understanding the Discriminant

The discriminant is the part of the quadratic formula under the square root: D = b² - 4ac. The discriminant determines the nature of the roots:

Discriminant (D) Nature of Roots Number of Roots
D > 0 Real and distinct 2
D = 0 Real and equal 1
D < 0 Complex conjugates 2

The discriminant provides important information about the behavior of the quadratic function and the nature of its roots.

Worked Examples

Example 1: Two Distinct Real Roots

Find the roots of x² - 5x + 6 = 0.

  1. Identify coefficients: a = 1, b = -5, c = 6.
  2. Calculate discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1.
  3. Since D > 0, there are two distinct real roots.
  4. Apply quadratic formula:

    x = [5 ± √1] / 2

    x₁ = (5 + 1)/2 = 3

    x₂ = (5 - 1)/2 = 2

Example 2: One Real Root

Find the root of x² - 6x + 9 = 0.

  1. Identify coefficients: a = 1, b = -6, c = 9.
  2. Calculate discriminant: D = (-6)² - 4(1)(9) = 36 - 36 = 0.
  3. Since D = 0, there is one real root.
  4. Apply quadratic formula:

    x = [6 ± √0] / 2 = 6/2 = 3

Example 3: Complex Roots

Find the roots of x² + 2x + 5 = 0.

  1. Identify coefficients: a = 1, b = 2, c = 5.
  2. Calculate discriminant: D = 2² - 4(1)(5) = 4 - 20 = -16.
  3. Since D < 0, the roots are complex conjugates.
  4. Apply quadratic formula:

    x = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2

    x₁ = -1 + 2i

    x₂ = -1 - 2i

FAQ

What is the difference between a linear and quadratic equation?

A linear equation has a single variable with the highest power of 1 (e.g., ax + b = 0), while a quadratic equation has a variable with the highest power of 2 (e.g., ax² + bx + c = 0).

Can a quadratic equation have no real roots?

Yes, if the discriminant is negative (D < 0), the quadratic equation has no real roots but two complex conjugate roots.

What is the significance of the discriminant?

The discriminant determines the nature of the roots: positive for two distinct real roots, zero for one real root, and negative for two complex conjugate roots.