If `x=(sqrt(a+2b)+sqrt(a-2b))/(sqrt(a+2b)-sqrt(a-2b))` then `bx^(2)+b` =

Options

A. ax
B. `0`
C. `-ax`
D. `-1`

Answer

To solve the problem given \( x = \frac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}} \) and find \( bx^2 + b \), we will follow these steps:

### Step 1: Rationalize the Denominator
We start with the expression for \( x \):
\[ x = \frac{\sqrt{a + 2b} + \sqrt{a - 2b}}{\sqrt{a + 2b} - \sqrt{a - 2b}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{(\sqrt{a + 2b} + \sqrt{a - 2b})(\sqrt{a + 2b} + \sqrt{a - 2b})}{(\sqrt{a + 2b} - \sqrt{a - 2b})(\sqrt{a + 2b} + \sqrt{a - 2b})} \]

### Step 2: Simplify the Denominator
The denominator simplifies using the difference of squares:
\[ (\sqrt{a + 2b})^2 - (\sqrt{a - 2b})^2 = (a + 2b) - (a - 2b) = 4b \]

### Step 3: Expand the Numerator
Now we expand the numerator:
\[ (\sqrt{a + 2b} + \sqrt{a - 2b})^2 = (\sqrt{a + 2b})^2 + 2\sqrt{(a + 2b)(a - 2b)} + (\sqrt{a - 2b})^2 \]
This simplifies to:
\[ (a + 2b) + (a - 2b) + 2\sqrt{(a + 2b)(a - 2b)} = 2a + 2\sqrt{(a + 2b)(a - 2b)} \]

### Step 4: Combine the Results
Now we can write \( x \):
\[ x = \frac{2a + 2\sqrt{(a + 2b)(a - 2b)}}{4b} = \frac{a + \sqrt{(a + 2b)(a - 2b)}}{2b} \]

### Step 5: Find \( x^2 \)
Next, we need to find \( x^2 \):
\[ x^2 = \left(\frac{a + \sqrt{(a + 2b)(a - 2b)}}{2b}\right)^2 = \frac{(a + \sqrt{(a + 2b)(a - 2b)})^2}{4b^2} \]

### Step 6: Expand \( x^2 \)
Expanding \( (a + \sqrt{(a + 2b)(a - 2b)})^2 \):
\[ x^2 = \frac{a^2 + 2a\sqrt{(a + 2b)(a - 2b)} + (a + 2b)(a - 2b)}{4b^2} \]
The term \( (a + 2b)(a - 2b) = a^2 - 4b^2 \), so:
\[ x^2 = \frac{a^2 + 2a\sqrt{(a + 2b)(a - 2b)} + a^2 - 4b^2}{4b^2} = \frac{2a^2 - 4b^2 + 2a\sqrt{(a + 2b)(a - 2b)}}{4b^2} \]

### Step 7: Calculate \( bx^2 + b \)
Now we compute \( bx^2 + b \):
\[ bx^2 + b = b\left(\frac{2a^2 - 4b^2 + 2a\sqrt{(a + 2b)(a - 2b)}}{4b^2}\right) + b \]
This simplifies to:
\[ = \frac{b(2a^2 - 4b^2 + 2a\sqrt{(a + 2b)(a - 2b)})}{4b^2} + b = \frac{2a^2 - 4b^2 + 2a\sqrt{(a + 2b)(a - 2b)}}{4b} + b \]

### Final Step: Combine Terms
Combining the terms gives us:
\[ bx^2 + b = \frac{2a^2 - 4b^2 + 2a\sqrt{(a + 2b)(a - 2b)} + 4b^2}{4b} = \frac{2a^2 + 2a\sqrt{(a + 2b)(a - 2b)}}{4b} \]

### Conclusion
Thus, the final result is:
\[ bx^2 + b = \frac{a(a + \sqrt{(a + 2b)(a - 2b)})}{2b} \]