If `(2p)/( p ^(2) - 2p +1) = (1)/(4),` then the value of `p + (1)/(p)` will be

Options

A. 8
B. 10
C. 12
D. None of these

Answer

To solve the equation \(\frac{2p}{p^2 - 2p + 1} = \frac{1}{4}\), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions:

\[ 2p \cdot 4 = 1 \cdot (p^2 - 2p + 1) \]

This simplifies to:

\[ 8p = p^2 - 2p + 1 \]

### Step 2: Rearrange the Equation
Next, we rearrange the equation to set it to zero:

\[ p^2 - 2p + 1 - 8p = 0 \]

Combining like terms gives:

\[ p^2 - 10p + 1 = 0 \]

### Step 3: Solve the Quadratic Equation
Now we will use the quadratic formula to solve for \(p\). The quadratic formula is:

\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation \(p^2 - 10p + 1 = 0\), we have \(a = 1\), \(b = -10\), and \(c = 1\). Plugging these values into the formula:

\[ p = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]

This simplifies to:

\[ p = \frac{10 \pm \sqrt{100 - 4}}{2} \]

\[ p = \frac{10 \pm \sqrt{96}}{2} \]

\[ p = \frac{10 \pm 4\sqrt{6}}{2} \]

\[ p = 5 \pm 2\sqrt{6} \]

### Step 4: Calculate \(p + \frac{1}{p}\)
Next, we need to find \(p + \frac{1}{p}\). We can use the identity:

\[ p + \frac{1}{p} = \frac{p^2 + 1}{p} \]

Calculating \(p^2\):

\[ p^2 = (5 \pm 2\sqrt{6})^2 = 25 + 20\sqrt{6} + 24 = 49 + 20\sqrt{6} \]

Now, substituting back into the identity:

\[ p + \frac{1}{p} = \frac{(49 + 20\sqrt{6}) + 1}{5 \pm 2\sqrt{6}} = \frac{50 + 20\sqrt{6}}{5 \pm 2\sqrt{6}} \]

To simplify this expression, we can evaluate it for both cases of \(p\) (i.e., \(5 + 2\sqrt{6}\) and \(5 - 2\sqrt{6}\)). However, we can also notice that since \(p + \frac{1}{p}\) is symmetric, we can directly compute it.

### Step 5: Conclusion
After evaluating both cases, we find that:

\[ p + \frac{1}{p} = 10 \]

Thus, the value of \(p + \frac{1}{p}\) is:

\[ \boxed{10} \]