## Determining the Value of 'k' for a Quadratic Equation with Equal Roots

A quadratic equation of the form *ax² + bx + c = 0* has equal roots when its discriminant, denoted by Δ, is equal to zero. The discriminant is calculated as:

**Δ = b² - 4ac**

In our given equation, *(k+4)x² + (k+1)x + 1 = 0*, we have:

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

To find the value of *k* for which the equation has equal roots, we set the discriminant to zero:

**(k+1)² - 4(k+4)(1) = 0**

Expanding and simplifying the equation:

- k² + 2k + 1 - 4k - 16 = 0
- k² - 2k - 15 = 0

Factoring the quadratic equation:

*(k - 5)(k + 3) = 0*

Therefore, the possible values of *k* are:

**k = 5****k = -3**

**Conclusion:**

We have determined that the quadratic equation *(k+4)x² + (k+1)x + 1 = 0* will have equal roots when **k = 5** or **k = -3**. This means that for these specific values of *k*, the equation will have only one distinct solution.