Determine the values of k for which the quadratic equation `(k + 4) x^(2) + (k + 1) x + 1 = 0` has equal roots.

Options

A. 5,2
B. `-3,3`
C. 2,4
D. 5, -3

Answer

To determine the values of \( k \) for which the quadratic equation

\[ (k + 4)x^2 + (k + 1)x + 1 = 0 \]

has equal roots, we need to use the condition for equal roots in a quadratic equation. A quadratic equation of the form \( ax^2 + bx + c = 0 \) has equal roots if its discriminant \( D \) is equal to zero. The discriminant is given by the formula:

\[ D = b^2 - 4ac \]

In our case, the coefficients are:
- \( a = k + 4 \)
- \( b = k + 1 \)
- \( c = 1 \)

Now, we can calculate the discriminant:

\[ D = (k + 1)^2 - 4(k + 4)(1) \]

Expanding this:

\[ D = (k + 1)^2 - 4(k + 4) \]

Calculating \( (k + 1)^2 \):

\[ (k + 1)^2 = k^2 + 2k + 1 \]

Now substituting back into the discriminant:

\[ D = k^2 + 2k + 1 - 4(k + 4) \]

Expanding \( -4(k + 4) \):

\[ -4(k + 4) = -4k - 16 \]

Now substituting this back into the discriminant:

\[ D = k^2 + 2k + 1 - 4k - 16 \]

Combining like terms:

\[ D = k^2 + (2k - 4k) + (1 - 16) \]
\[ D = k^2 - 2k - 15 \]

Now, we set the discriminant equal to zero for equal roots:

\[ k^2 - 2k - 15 = 0 \]

Next, we can factor this quadratic equation:

\[ (k - 5)(k + 3) = 0 \]

Setting each factor equal to zero gives us the possible values for \( k \):

1. \( k - 5 = 0 \) → \( k = 5 \)
2. \( k + 3 = 0 \) → \( k = -3 \)

Thus, the values of \( k \) for which the quadratic equation has equal roots are:

\[ \boxed{5 \text{ and } -3} \]