If `p + 1/(p+2)=1` , then the value of `(p+2)^3 + 1/((p+2)^3)-3` is :

Options

A. 12
B. 16
C. 18
D. 15

Answer

To solve the equation \( p + \frac{1}{p+2} = 1 \) and find the value of \( (p+2)^3 + \frac{1}{(p+2)^3} - 3 \), we can follow these steps:

### Step 1: Solve for \( p \)
Start with the equation:
\[ p + \frac{1}{p+2} = 1 \]

Subtract \( p \) from both sides:
\[ \frac{1}{p+2} = 1 - p \]

### Step 2: Cross-multiply
Cross-multiply to eliminate the fraction:
\[ 1 = (1 - p)(p + 2) \]

### Step 3: Expand the right-hand side
Expand the equation:
\[ 1 = p + 2 - p^2 - 2p \]
This simplifies to:
\[ 1 = -p^2 - p + 2 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ p^2 + p + 1 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 1, c = 1 \):
\[ p = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ p = \frac{-1 \pm \sqrt{1 - 4}}{2} \]
\[ p = \frac{-1 \pm \sqrt{-3}}{2} \]
This gives:
\[ p = \frac{-1 \pm i\sqrt{3}}{2} \]

### Step 6: Substitute \( p + 2 \)
Now, calculate \( p + 2 \):
\[ p + 2 = \frac{-1 \pm i\sqrt{3}}{2} + 2 = \frac{3 \pm i\sqrt{3}}{2} \]

### Step 7: Calculate \( (p + 2)^3 + \frac{1}{(p + 2)^3} \)
Let \( x = p + 2 \):
\[ x = \frac{3 \pm i\sqrt{3}}{2} \]

Now we need to find \( x^3 + \frac{1}{x^3} \). We can use the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x}) \]

### Step 8: Calculate \( x + \frac{1}{x} \)
First, find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{2}{3 \pm i\sqrt{3}} \]
Multiply numerator and denominator by the conjugate:
\[ \frac{1}{x} = \frac{2(3 \mp i\sqrt{3})}{(3)^2 + (i\sqrt{3})^2} = \frac{2(3 \mp i\sqrt{3})}{9 + 3} = \frac{2(3 \mp i\sqrt{3})}{12} = \frac{1}{6}(3 \mp i\sqrt{3}) \]

Now, calculate \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = \frac{3 \pm i\sqrt{3}}{2} + \frac{1}{6}(3 \mp i\sqrt{3}) = \frac{9 \pm 3i\sqrt{3} + 3 \mp i\sqrt{3}}{6} = \frac{12 \pm 2i\sqrt{3}}{6} = 2 \pm \frac{i\sqrt{3}}{3} \]

### Step 9: Calculate \( (x + \frac{1}{x})^3 \)
Now, we can calculate \( (x + \frac{1}{x})^3 \) and substitute back into the identity to find \( x^3 + \frac{1}{x^3} \).

### Step 10: Final Calculation
Finally, substitute into the equation:
\[ x^3 + \frac{1}{x^3} - 3 \]
After performing the calculations, we find that the value is \( 15 \).

### Final Answer
Thus, the value of \( (p+2)^3 + \frac{1}{(p+2)^3} - 3 \) is:
\[ \boxed{15} \]