If `(x+a)^(2) = x^(2) + 1 + (1)/( 4x^2)`, then find `a.`

Options

A. `a = (1)/( 5x)`
B. `a = (1)/( 2x)`
C. `a = (1)/( 3x)`
D. `a = (1)/( 4x)`

Answer

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

### Step 1: Expand the left-hand side
Using the formula for the square of a binomial, we have:
\[ (x + a)^2 = x^2 + 2ax + a^2 \]

### Step 2: Set the expanded form equal to the right-hand side
Now we can set the expanded left-hand side equal to the right-hand side:
\[ x^2 + 2ax + a^2 = x^2 + 1 + \frac{1}{4x^2} \]

### Step 3: Cancel \(x^2\) from both sides
Since \(x^2\) appears on both sides, we can cancel it:
\[ 2ax + a^2 = 1 + \frac{1}{4x^2} \]

### Step 4: Rearrange the equation
Now we need to isolate terms involving \(a\):
\[ 2ax + a^2 - 1 - \frac{1}{4x^2} = 0 \]

### Step 5: Identify coefficients
This is a quadratic equation in terms of \(a\). We can rewrite it as:
\[ a^2 + 2ax - 1 - \frac{1}{4x^2} = 0 \]

### Step 6: Use the quadratic formula
To find \(a\), we can use the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(b = 2x\), \(c = -\left(1 + \frac{1}{4x^2}\right)\).

### Step 7: Calculate the discriminant
Calculate the discriminant:
\[ b^2 - 4ac = (2x)^2 - 4 \cdot 1 \cdot \left(-\left(1 + \frac{1}{4x^2}\right)\right) \]
\[ = 4x^2 + 4\left(1 + \frac{1}{4x^2}\right) \]
\[ = 4x^2 + 4 + 1 = 4x^2 + 5 \]

### Step 8: Substitute back into the quadratic formula
Now substitute back into the quadratic formula:
\[ a = \frac{-2x \pm \sqrt{4x^2 + 5}}{2} \]
\[ = -x \pm \frac{\sqrt{4x^2 + 5}}{2} \]

### Step 9: Simplify the expression
Thus, we have two possible values for \(a\):
\[ a = -x + \frac{\sqrt{4x^2 + 5}}{2} \quad \text{or} \quad a = -x - \frac{\sqrt{4x^2 + 5}}{2} \]

### Conclusion
The value of \(a\) depends on the value of \(x\). However, if we need a specific value for \(a\), we can evaluate it for a particular value of \(x\).