If `2m - 1/(2m)=2`, where `mne0`, then the value of `m^2+1/(16m^2)` is

Options

A. `2 1/2`
B. 2
C. `1 1/2`
D. 4

Answer

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

### Step 1: Solve the equation for \( m \)

Starting with the equation:

\[ 2m - \frac{1}{2m} = 2 \]

We can first eliminate the fraction by multiplying both sides by \( 2m \) (noting that \( m \neq 0 \)):

\[ 2m(2m) - 1 = 2(2m) \]

This simplifies to:

\[ 4m^2 - 1 = 4m \]

### Step 2: Rearrange the equation

Now, we can rearrange the equation to form a standard quadratic equation:

\[ 4m^2 - 4m - 1 = 0 \]

### Step 3: Use the quadratic formula

To solve for \( m \), we can use the quadratic formula:

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

Here, \( a = 4 \), \( b = -4 \), and \( c = -1 \). Plugging in these values:

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

Calculating the discriminant:

\[ m = \frac{4 \pm \sqrt{16 + 16}}{8} \]
\[ m = \frac{4 \pm \sqrt{32}}{8} \]
\[ m = \frac{4 \pm 4\sqrt{2}}{8} \]
\[ m = \frac{1 \pm \sqrt{2}}{2} \]

### Step 4: Find \( m^2 + \frac{1}{16m^2} \)

Now we need to calculate \( m^2 + \frac{1}{16m^2} \). First, we can find \( m^2 \):

\[ m^2 = \left(\frac{1 \pm \sqrt{2}}{2}\right)^2 = \frac{(1 \pm \sqrt{2})^2}{4} = \frac{1 + 2 \pm 2\sqrt{2}}{4} = \frac{3 \pm 2\sqrt{2}}{4} \]

Next, we calculate \( \frac{1}{16m^2} \):

\[ \frac{1}{16m^2} = \frac{1}{16 \cdot \frac{3 \pm 2\sqrt{2}}{4}} = \frac{1}{4(3 \pm 2\sqrt{2})} = \frac{1}{12 \pm 8\sqrt{2}} \]

To simplify the addition \( m^2 + \frac{1}{16m^2} \):

\[ m^2 + \frac{1}{16m^2} = \frac{3 \pm 2\sqrt{2}}{4} + \frac{1}{12 \pm 8\sqrt{2}} \]

This requires a common denominator, which can be complex. Instead, we can use the identity:

\[ m^2 + \frac{1}{16m^2} = \left(m - \frac{1}{4m}\right)^2 + \frac{1}{16} \]

From our earlier steps, we know:

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

### Final Calculation

After simplifying, we find that:

\[ m^2 + \frac{1}{16m^2} = \frac{3}{2} \]

Thus, the final answer is:

\[ \frac{3}{2} \text{ or } 1 \frac{1}{2} \]