If `a=x+sqrt(x^(2)+1)` , then what is x equal to?

Options

A. `(1//2)(a+a^(-1))`
B. ` (1//2)(a-a^(-1))`
C. `a+a^(-1)`
D. ` a-a^(-1)`

Explanation

To solve the equation \( a = x + \sqrt{x^2 + 1} \) for \( x \), we can follow these steps:

### Step 1: Isolate the square root
Start with the original equation:
\[ a = x + \sqrt{x^2 + 1} \]
Subtract \( x \) from both sides:
\[ a - x = \sqrt{x^2 + 1} \]

### Step 2: Square both sides
Square both sides to eliminate the square root:
\[ (a - x)^2 = x^2 + 1 \]

### Step 3: Expand the left side
Expand the left side:
\[ a^2 - 2ax + x^2 = x^2 + 1 \]

### Step 4: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ a^2 - 2ax = 1 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ a^2 - 1 = 2ax \]

### Step 6: Solve for \( x \)
Now, divide both sides by \( 2a \) (assuming \( a \neq 0 \)):
\[ x = \frac{a^2 - 1}{2a} \]

### Final Result
Thus, the value of \( x \) is:
\[ x = \frac{a^2 - 1}{2a} \]

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