Number of Real Solutions in A Quadratic Equation Calculator

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2. It has the general form:

Where:

• a, b, and c are constants
• a ≠ 0 (if a = 0, the equation becomes linear)
• x represents the variable

Quadratic equations can be solved using factoring, completing the square, or the quadratic formula. The number of real solutions depends on the discriminant.

How to Find Real Solutions

The number of real solutions to a quadratic equation can be determined by analyzing the discriminant. The discriminant is a part of the quadratic formula that tells us the nature of the roots.

The relationship between the discriminant and the number of real solutions is as follows:

• If D > 0: Two distinct real solutions
• If D = 0: One real solution (a repeated root)
• If D < 0: No real solutions (the solutions are complex)

Using the Discriminant

The discriminant provides important information about the roots of the quadratic equation:

1. Calculate the discriminant using D = b² - 4ac
2. Compare the discriminant to zero:
   • Positive discriminant: Two real solutions
   • Zero discriminant: One real solution
   • Negative discriminant: No real solutions

This method is efficient and can be applied to any quadratic equation to determine the number of real solutions without solving the equation completely.

Examples

Example 1: Two Real Solutions

Consider the equation x² - 5x + 6 = 0

• a = 1, b = -5, c = 6
• Discriminant D = (-5)² - 4(1)(6) = 25 - 24 = 1
• Since D > 0, there are two real solutions

Example 2: One Real Solution

Consider the equation x² - 6x + 9 = 0

• a = 1, b = -6, c = 9
• Discriminant D = (-6)² - 4(1)(9) = 36 - 36 = 0
• Since D = 0, there is one real solution

Example 3: No Real Solutions

Consider the equation x² + 2x + 5 = 0

• a = 1, b = 2, c = 5
• Discriminant D = (2)² - 4(1)(5) = 4 - 20 = -16
• Since D < 0, there are no real solutions