If ` 2 p + (1)/( p) = 4` then value of ` p^(3) + (1)/(8 p ^(3))` is

Options

A. 4
B. 5
C. 8
D. 15

Answer

To solve the equation \( 2p + \frac{1}{p} = 4 \) and find the value of \( p^3 + \frac{1}{8p^3} \), we can follow these steps:

### Step 1: Solve for \( p \)
We start with the equation:
\[ 2p + \frac{1}{p} = 4 \]
To eliminate the fraction, we can multiply both sides by \( p \):
\[ 2p^2 + 1 = 4p \]
Rearranging the equation gives us:
\[ 2p^2 - 4p + 1 = 0 \]

### Step 2: Use the quadratic formula
We can solve the quadratic equation \( 2p^2 - 4p + 1 = 0 \) using the quadratic formula:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -4 \), and \( c = 1 \):
\[ p = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} \]
Calculating the discriminant:
\[ p = \frac{4 \pm \sqrt{16 - 8}}{4} = \frac{4 \pm \sqrt{8}}{4} = \frac{4 \pm 2\sqrt{2}}{4} = 1 \pm \frac{\sqrt{2}}{2} \]

### Step 3: Calculate \( p^3 + \frac{1}{8p^3} \)
Now, we need to find \( p^3 + \frac{1}{8p^3} \). We can use the identity for cubes:
\[ p^3 + \frac{1}{p^3} = (p + \frac{1}{p})^3 - 3(p + \frac{1}{p}) \]
First, we find \( p + \frac{1}{p} \):
From the original equation \( 2p + \frac{1}{p} = 4 \), we can express \( p + \frac{1}{p} \):
\[ p + \frac{1}{p} = 2 \]

Now we can calculate \( p^3 + \frac{1}{p^3} \):
\[ p^3 + \frac{1}{p^3} = (2)^3 - 3(2) = 8 - 6 = 2 \]

Next, we need to find \( \frac{1}{8p^3} \):
Since \( p^3 + \frac{1}{p^3} = 2 \), we can express \( \frac{1}{8p^3} \) as:
\[ \frac{1}{8p^3} = \frac{1}{8} \cdot \frac{1}{p^3} \]
We already have \( p^3 \) from our earlier calculation.

### Step 4: Combine the results
Thus, we can find:
\[ p^3 + \frac{1}{8p^3} = 2 + \frac{1}{8} \cdot \frac{1}{p^3} \]
To find \( \frac{1}{p^3} \), we can use the earlier result \( p^3 + \frac{1}{p^3} = 2 \) to find \( p^3 \) and then substitute back to find the final answer.

### Final Calculation
Since \( p^3 + \frac{1}{p^3} = 2 \), we can find:
\[ p^3 + \frac{1}{8p^3} = 2 + \frac{1}{8} \cdot \frac{1}{p^3} = 2 + \frac{1}{8} \cdot (2 - p^3) \]
This leads us to:
\[ p^3 + \frac{1}{8p^3} = 2 + \frac{1}{8} \cdot (2) = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \]

Thus, the final answer is:
\[ \frac{9}{4} \]