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Roots of Quadratic Equation in A Bi Form Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the roots of a quadratic equation in the form ax² + bx + c = 0 using the quadratic formula. The roots are the values of x that satisfy the equation. The calculator provides both real and complex roots when applicable.

Introduction

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

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The roots of the equation are the values of x that satisfy the equation. There are three possible cases for the roots:

  • Two distinct real roots
  • One real root (a repeated root)
  • Two complex conjugate roots

The roots can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

This calculator implements this formula to find the roots of any quadratic equation you provide.

Formula

The quadratic formula is derived from completing the square and solving for x. The formula is:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • a, b, c are coefficients of the quadratic equation
  • √(b² - 4ac) is the discriminant
  • The ± symbol indicates there are two roots

The discriminant (b² - 4ac) determines the nature of the roots:

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

How to Use the Calculator

  1. Enter the coefficients a, b, and c of your quadratic equation in the form ax² + bx + c = 0
  2. Click the "Calculate" button
  3. View the results showing the roots of the equation
  4. Interpret the results based on the discriminant

Note: The coefficient a must not be zero. If a is zero, the equation is no longer quadratic and the calculator will show an error.

Interpreting Results

The calculator provides the roots of the quadratic equation. The interpretation depends on the discriminant:

  • Two distinct real roots: The equation crosses the x-axis at two different points. The parabola opens upwards if a > 0 or downwards if a < 0.
  • One real root (repeated): The equation touches the x-axis at exactly one point. The parabola has a vertex on the x-axis.
  • Two complex conjugate roots: The equation does not cross the x-axis in the real plane. The roots are complex numbers.

The calculator also shows the discriminant value and its interpretation.

Worked Examples

Example 1: Two distinct real roots

Find the roots of x² - 5x + 6 = 0.

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 x = [5 ± √1] / 2 x = [5 ± 1] / 2

The roots are x = 3 and x = 2.

Example 2: One real root (repeated)

Find the roots of x² - 6x + 9 = 0.

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 x = [6 ± √0] / 2 x = 6 / 2 = 3

The root is x = 3 (repeated).

Example 3: Two complex conjugate roots

Find the roots of x² + 2x + 5 = 0.

Using the quadratic formula:

x = [-2 ± √(4 - 20)] / 2 x = [-2 ± √(-16)] / 2 x = [-2 ± 4i] / 2

The roots are x = -1 + 2i and x = -1 - 2i.

Frequently Asked Questions

What is the quadratic formula?
The quadratic formula is a method for solving quadratic equations of the form ax² + bx + c = 0. It is given by x = [-b ± √(b² - 4ac)] / (2a).
What does the discriminant tell me?
The discriminant (b² - 4ac) determines the nature of the roots. A positive discriminant indicates two distinct real roots, zero indicates one real root, and a negative discriminant indicates two complex conjugate roots.
Can the quadratic formula be used for any quadratic equation?
Yes, the quadratic formula can be used for any quadratic equation as long as the coefficient a is not zero. If a is zero, the equation is no longer quadratic.
What are complex roots?
Complex roots are roots that involve the imaginary unit i, where i² = -1. They occur when the discriminant is negative.