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Nature of The Roots Find Coefficients Calculator

Reviewed by Calculator Editorial Team

This calculator helps you determine the nature of the roots (real, complex, or repeated) of a quadratic equation and find the corresponding coefficients. Understanding the nature of roots is essential in algebra and has practical applications in physics, engineering, and economics.

Introduction

A quadratic equation is a second-degree polynomial equation of the form:

ax² + bx + c = 0

The nature of the roots of a quadratic equation can be determined by analyzing its discriminant, which is given by:

D = b² - 4ac

The discriminant provides important information about the roots:

  • If D > 0, the equation has two distinct real roots.
  • If D = 0, the equation has exactly one real root (a repeated root).
  • If D < 0, the equation has two complex conjugate roots.

This calculator allows you to input the coefficients a, b, and c and determines the nature of the roots along with the actual roots themselves.

How to Use This Calculator

  1. Enter the coefficient a (the coefficient of x²).
  2. Enter the coefficient b (the coefficient of x).
  3. Enter the constant term c.
  4. Click the "Calculate" button to determine the nature of the roots and find the roots.
  5. Review the results, which will show the discriminant, nature of the roots, and the roots themselves.

Note: The coefficient a cannot be zero as it would no longer be a quadratic equation.

Formula

The roots of the quadratic equation ax² + bx + c = 0 can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, c are the coefficients of the quadratic equation
  • √(b² - 4ac) is the square root of the discriminant

The discriminant D = b² - 4ac determines the nature of the roots:

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

Worked Example

Let's find the roots of the equation 2x² - 4x - 6 = 0.

  1. Identify the coefficients: a = 2, b = -4, c = -6
  2. Calculate the discriminant: D = (-4)² - 4(2)(-6) = 16 + 48 = 64
  3. Since D > 0, there are two distinct real roots.
  4. Apply the quadratic formula:
    x = [4 ± √64] / 4 = [4 ± 8] / 4
  5. Calculate the two roots:
    • First root: (4 + 8)/4 = 12/4 = 3
    • Second root: (4 - 8)/4 = -4/4 = -1

The roots of the equation are x = 3 and x = -1.

Interpreting the Results

When you use this calculator, you'll receive several key pieces of information:

  1. Discriminant (D): This tells you the nature of the roots:
    • D > 0: Two distinct real roots
    • D = 0: One real root (repeated)
    • D < 0: Two complex conjugate roots
  2. Roots: The actual solutions to the quadratic equation.

Understanding these results helps you analyze the behavior of the quadratic function and its graph.

FAQ

What is the discriminant in a quadratic equation?

The discriminant is the part of the quadratic formula under the square root (b² - 4ac). It determines the nature of the roots of the quadratic equation.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant is greater than or equal to zero (D ≥ 0).

What does it mean if the discriminant is negative?

A negative discriminant means the quadratic equation has two complex conjugate roots, which are not real numbers.

Can a quadratic equation have only one root?

Yes, a quadratic equation has exactly one real root (a repeated root) when the discriminant is zero (D = 0).