(k+1)x^2+2(k+3)x+(k+8)=0

4 min read Jun 16, 2024

Analyzing the Quadratic Equation: (k+1)x^2 + 2(k+3)x + (k+8) = 0

This article explores the quadratic equation (k+1)x^2 + 2(k+3)x + (k+8) = 0, examining its properties and how the value of k affects its solutions.

Understanding the Basics

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation are known as its roots. These roots can be real or complex numbers.

The Discriminant and Nature of Roots

The discriminant, represented by Δ, determines the nature of the roots of a quadratic equation. It is calculated as follows:

Δ = b² - 4ac

For the equation (k+1)x² + 2(k+3)x + (k+8) = 0, the discriminant is:

Δ = [2(k+3)]² - 4(k+1)(k+8)

Let's analyze the different scenarios based on the value of Δ:

Δ > 0: The equation has two distinct real roots.
Δ = 0: The equation has one real root (a double root).
Δ < 0: The equation has two complex roots (conjugate pairs).

Finding the Roots

The roots of the quadratic equation can be found using the quadratic formula:

x = (-b ± √Δ) / 2a

For our equation, the quadratic formula becomes:

x = [-2(k+3) ± √[4(k+3)² - 4(k+1)(k+8)]] / 2(k+1)

x = [-(k+3) ± √(k² + 2k - 8)] / (k+1)

This formula provides the two roots of the equation for any given value of k.

Impact of 'k' on Solutions

The value of k directly impacts the discriminant and, consequently, the nature and value of the roots. We can analyze the different possibilities:

k = -1: This leads to a degenerate case, where the equation reduces to a linear equation. It will have a single real root, but it's not a true quadratic solution.
k = -8: The equation simplifies to a linear equation.
k > -1 and k ≠ -8: The equation will have two distinct real roots, as the discriminant is positive in this range.
-8 < k < -1: The equation will have two complex roots, as the discriminant is negative in this range.

Conclusion

By understanding the discriminant, the quadratic formula, and the impact of the parameter k, we can gain valuable insights into the solutions of the quadratic equation (k+1)x² + 2(k+3)x + (k+8) = 0. These insights enable us to analyze the equation's behavior across different values of k and determine the nature and characteristics of its roots.