## Analyzing the Quadratic Equation: (k+4)x² + (k+1)x + 1 = 0

This article explores the quadratic equation **(k+4)x² + (k+1)x + 1 = 0**, focusing on its properties and how the value of 'k' influences its solutions.

### Understanding the Equation

The given equation is a quadratic equation in the form **ax² + bx + c = 0**, where:

**a = (k+4)****b = (k+1)****c = 1**

The value of 'k' is a constant that can be any real number, influencing the nature of the solutions to the quadratic equation.

### Finding the Solutions

The solutions to a quadratic equation can be found using the quadratic formula:

**x = [-b ± √(b² - 4ac)] / 2a**

Substituting the values from our equation:

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

Simplifying this expression will give us the two solutions for 'x' in terms of 'k'.

### Analyzing the Discriminant

The **discriminant** of a quadratic equation, represented by **Δ = b² - 4ac**, provides valuable information about the nature of the solutions.

**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.

For our equation, the discriminant is:

**Δ = (k+1)² - 4(k+4)(1) = k² - 6k - 15**

By analyzing this expression, we can determine the values of 'k' that lead to different types of solutions:

**k² - 6k - 15 > 0:**The equation will have**two distinct real roots**.**k² - 6k - 15 = 0:**The equation will have**one real root (a double root)**.**k² - 6k - 15 < 0:**The equation will have**two complex roots**.

### Conclusion

The solutions to the quadratic equation **(k+4)x² + (k+1)x + 1 = 0** are dependent on the value of 'k'. By analyzing the discriminant, we can understand the nature of the solutions and determine the values of 'k' that lead to real or complex roots.

This understanding is crucial in various mathematical and scientific applications where quadratic equations are used to model real-world phenomena.