Find the value of k if the equation `(k+1)x^(2)+2(k+3)x+(k+8)=0` as have equal roots.

Answer

To find the value of \( k \) for which the quadratic equation

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

has equal roots, we will follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = k + 1 \)
- \( b = 2(k + 3) = 2k + 6 \)
- \( c = k + 8 \)

### Step 2: Use the condition for equal roots
For a quadratic equation to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by:

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

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = (2k + 6)^2 - 4(k + 1)(k + 8) \]

### Step 3: Expand the discriminant
Now, we will expand both parts of the discriminant:

1. Expanding \( (2k + 6)^2 \):

\[ (2k + 6)^2 = 4k^2 + 24k + 36 \]

2. Expanding \( 4(k + 1)(k + 8) \):

\[ 4(k + 1)(k + 8) = 4(k^2 + 9k + 8) = 4k^2 + 36k + 32 \]

### Step 4: Set the discriminant to zero
Now, substituting back into the discriminant:

\[ 4k^2 + 24k + 36 - (4k^2 + 36k + 32) = 0 \]

### Step 5: Simplify the equation
Now, simplify the equation:

\[ 4k^2 + 24k + 36 - 4k^2 - 36k - 32 = 0 \]

This simplifies to:

\[ -12k + 4 = 0 \]

### Step 6: Solve for \( k \)
Now, solve for \( k \):

\[ -12k + 4 = 0 \implies 12k = 4 \implies k = \frac{4}{12} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( k \) for which the equation has equal roots is

\[ \boxed{\frac{1}{3}} \]