If `4 x^2 = sqrt5 + 2,` then the value of `4 x^2 - frac(1) (4 x^2 ) is

Options

A. 1
B. -1
C. 16
D. 4

Answer

To solve the equation \( 4x^2 = \sqrt{5} + 2 \) and find the value of \( 4x^2 - \frac{1}{4x^2} \), we will follow these steps:

### Step 1: Substitute the value of \( 4x^2 \)
We know from the problem statement that:
\[ 4x^2 = \sqrt{5} + 2 \]

### Step 2: Calculate \( \frac{1}{4x^2} \)
To find \( \frac{1}{4x^2} \), we take the reciprocal of \( 4x^2 \):
\[ \frac{1}{4x^2} = \frac{1}{\sqrt{5} + 2} \]

### Step 3: Rationalize the denominator
To simplify \( \frac{1}{\sqrt{5} + 2} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{5} - 2 \):
\[ \frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{\sqrt{5} - 2}{(\sqrt{5})^2 - 2^2} \]
Calculating the denominator:
\[ (\sqrt{5})^2 - 2^2 = 5 - 4 = 1 \]
Thus, we have:
\[ \frac{1}{4x^2} = \sqrt{5} - 2 \]

### Step 4: Substitute back into the expression
Now we substitute \( 4x^2 \) and \( \frac{1}{4x^2} \) back into the expression \( 4x^2 - \frac{1}{4x^2} \):
\[ 4x^2 - \frac{1}{4x^2} = (\sqrt{5} + 2) - (\sqrt{5} - 2) \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ (\sqrt{5} + 2) - (\sqrt{5} - 2) = \sqrt{5} + 2 - \sqrt{5} + 2 = 4 \]

### Final Answer
Thus, the value of \( 4x^2 - \frac{1}{4x^2} \) is:
\[ \boxed{4} \]