(k-5)x^2+2(k-5)x+2=0

4 min read Jun 16, 2024

Analyzing the Quadratic Equation: (k-5)x² + 2(k-5)x + 2 = 0

This article will delve into the quadratic equation (k-5)x² + 2(k-5)x + 2 = 0, exploring its properties, solutions, and the impact of the parameter 'k'.

Understanding the Basics

The equation is a quadratic equation because the highest power of the variable 'x' is 2. The coefficients of the equation are dependent on the parameter 'k'.

Key Concepts:

Discriminant: The discriminant of a quadratic equation ax² + bx + c = 0 is given by Δ = b² - 4ac. This value tells us about the nature of the roots (solutions) of the equation:
- Δ > 0: Two distinct real roots.
- Δ = 0: One real root (a double root).
- Δ < 0: Two complex roots.
Roots: The roots of a quadratic equation are the values of 'x' that satisfy the equation.

Finding the Discriminant

For our equation, (k-5)x² + 2(k-5)x + 2 = 0:

a = (k-5)
b = 2(k-5)
c = 2

Therefore, the discriminant Δ is:

Δ = (2(k-5))² - 4(k-5)(2) = 4(k-5)² - 8(k-5)

Analyzing the Roots

Let's analyze the nature of the roots based on the discriminant:

Δ > 0:
- This means 4(k-5)² - 8(k-5) > 0.
- Simplifying, we get 4(k-5)(k-7) > 0.
- This inequality holds true when k < 5 or k > 7.
- Conclusion: For k < 5 or k > 7, the equation has two distinct real roots.
Δ = 0:
- This means 4(k-5)² - 8(k-5) = 0.
- Solving for 'k', we get k = 5 or k = 7.
- Conclusion: For k = 5 or k = 7, the equation has one real root (a double root).
Δ < 0:
- This means 4(k-5)² - 8(k-5) < 0.
- Simplifying, we get 4(k-5)(k-7) < 0.
- This inequality holds true when 5 < k < 7.
- Conclusion: For 5 < k < 7, the equation has two complex roots.

Conclusion

The quadratic equation (k-5)x² + 2(k-5)x + 2 = 0 exhibits diverse behavior depending on the value of 'k'. The nature of the roots can be determined by analyzing the discriminant, which allows us to understand the characteristics of the solutions for different values of the parameter.