What is the Nature of the Roots of the Quadratic Equation 2x² - 8x + 5 = 0?

The roots of the quadratic equation 2x² - 8x + 5 = 0 are real and unequal.

Understanding the Nature of Quadratic Roots

The nature of the roots of any quadratic equation in the standard form ax² + bx + c = 0 (where a ≠ 0) is determined by its discriminant, denoted by the Greek letter delta (Δ). This crucial value indicates whether the roots are real or complex, and if real, whether they are distinct or identical. The formula for the discriminant is:

Δ = b² - 4ac

By evaluating the discriminant, we can classify the roots into distinct categories, which is fundamental in algebra for predicting the type of solutions a quadratic equation will yield.

How to Determine Root Nature Using the Discriminant

To determine the nature of the roots for a given quadratic equation, follow these straightforward steps:

1. Identify Coefficients: Begin by comparing the given quadratic equation to the standard form ax² + bx + c = 0 to accurately identify the numerical values of a, b, and c. Ensure you include any negative signs with the coefficients.
2. Calculate the Discriminant: Substitute the identified values of a, b, and c into the discriminant formula: Δ = b² - 4ac. Perform the arithmetic carefully to find the value of Δ.
3. Interpret the Result: Analyze the calculated value of Δ to determine the nature of the roots:
   • If Δ > 0: The equation has two distinct real roots. This means there are two different real number solutions for x.
   • If Δ = 0: The equation has exactly one real root, which is often referred to as a repeated root or two equal real roots.
   • If Δ < 0: The equation has two distinct complex (non-real) roots. These roots are always conjugates of each other and involve the imaginary unit i.

Applying to 2x² - 8x + 5 = 0

Let's apply this methodical approach to the specific equation 2x² - 8x + 5 = 0:

• Step 1: Identify Coefficients
  Comparing 2x² - 8x + 5 = 0 with ax² + bx + c = 0:

  • a = 2
  • b = -8
  • c = 5

• Step 2: Calculate the Discriminant
  Substitute these values into the discriminant formula:
  Δ = b² - 4ac
  Δ = (-8)² - 4(2)(5)
  Δ = 64 - 40
  Δ = 24

• Step 3: Interpret the Result
  Since the calculated discriminant Δ = 24, which is greater than 0 (Δ > 0), we conclude that the quadratic equation 2x² - 8x + 5 = 0 has real and unequal roots. This means there are two distinct real number solutions for x.

Summary of Discriminant and Root Nature

For a clear and quick reference, the table below summarizes the relationship between the discriminant's value and the corresponding nature of the quadratic roots:

This comprehensive analysis confirms that for the given equation, the positive discriminant value directly indicates real and unequal roots, allowing for two unique real number solutions to the quadratic equation.