If `2a - (1)/(2a) = 3,` then `16 a ^(4) + (1)/(16 a ^(4))` is equal to

Options

A. 11
B. 119
C. 117
D. 121

Answer

To solve the equation \(2a - \frac{1}{2a} = 3\) and find the value of \(16a^4 + \frac{1}{16a^4}\), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ 2a - \frac{1}{2a} = 3 \]

### Step 2: Rearrange the equation
Add \(\frac{1}{2a}\) to both sides:
\[ 2a = 3 + \frac{1}{2a} \]

### Step 3: Multiply both sides by \(2a\) to eliminate the fraction
\[ 2a \cdot 2a = (3 + \frac{1}{2a}) \cdot 2a \]
This simplifies to:
\[ 4a^2 = 6a + 1 \]

### Step 4: Rearrange to form a quadratic equation
Rearranging gives:
\[ 4a^2 - 6a - 1 = 0 \]

### Step 5: Use the quadratic formula to solve for \(a\)
The quadratic formula is given by:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 4\), \(b = -6\), and \(c = -1\):
\[ a = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4} \]
Calculating the discriminant:
\[ b^2 - 4ac = 36 + 16 = 52 \]
Thus,
\[ a = \frac{6 \pm \sqrt{52}}{8} = \frac{6 \pm 2\sqrt{13}}{8} = \frac{3 \pm \sqrt{13}}{4} \]

### Step 6: Find \(2a\)
Now, we can find \(2a\):
\[ 2a = \frac{3 \pm \sqrt{13}}{2} \]

### Step 7: Calculate \(16a^4 + \frac{1}{16a^4}\)
To find \(16a^4 + \frac{1}{16a^4}\), we can use the identity:
\[ x = 2a \implies x - \frac{1}{x} = 3 \]
Squaring both sides:
\[ \left(x - \frac{1}{x}\right)^2 = 9 \implies x^2 - 2 + \frac{1}{x^2} = 9 \]
Thus,
\[ x^2 + \frac{1}{x^2} = 11 \]

### Step 8: Square again to find \(16a^4 + \frac{1}{16a^4}\)
Now, square \(x^2 + \frac{1}{x^2}\):
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 121 \implies x^4 + 2 + \frac{1}{x^4} = 121 \]
Thus,
\[ x^4 + \frac{1}{x^4} = 119 \]

### Step 9: Substitute back to find \(16a^4 + \frac{1}{16a^4}\)
Since \(x = 2a\):
\[ 16a^4 + \frac{1}{16a^4} = x^4 + \frac{1}{x^4} = 119 \]

### Final Answer
Therefore, the value of \(16a^4 + \frac{1}{16a^4}\) is:
\[ \boxed{119} \]