If `x=sqrt((2+sqrt(3))/(2-sqrt(3)))`, then what is the value of `(x^(2)+x-9)` ?

Options

A. `0`
B. `3sqrt(2)`
C. `3sqrt(3)`
D. `5sqrt(3)`

Answer

To solve the problem, we start with the expression given for \( x \):

\[ x = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}} \]

### Step 1: Simplify \( x \)

To simplify \( x \), we will multiply the numerator and the denominator by the conjugate of the denominator, which is \( 2 + \sqrt{3} \):

\[ x = \sqrt{\frac{(2 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})}} \]

### Step 2: Calculate the denominator

Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):

\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 3: Calculate the numerator

Now, we calculate the numerator:

\[ (2 + \sqrt{3})(2 + \sqrt{3}) = (2 + \sqrt{3})^2 = 2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 4: Substitute back into \( x \)

Now substituting back, we have:

\[ x = \sqrt{7 + 4\sqrt{3}} \]

### Step 5: Calculate \( x^2 \)

Next, we calculate \( x^2 \):

\[ x^2 = 7 + 4\sqrt{3} \]

### Step 6: Substitute \( x \) and \( x^2 \) into \( x^2 + x - 9 \)

Now we substitute \( x^2 \) and \( x \) into the expression \( x^2 + x - 9 \):

\[ x^2 + x - 9 = (7 + 4\sqrt{3}) + \sqrt{7 + 4\sqrt{3}} - 9 \]

### Step 7: Simplify the expression

This simplifies to:

\[ (7 - 9) + 4\sqrt{3} + \sqrt{7 + 4\sqrt{3}} = -2 + 4\sqrt{3} + \sqrt{7 + 4\sqrt{3}} \]

### Step 8: Final simplification

To simplify further, we can calculate \( \sqrt{7 + 4\sqrt{3}} \) directly or leave it as is. However, we can see that we have:

\[ x^2 + x - 9 = -2 + 4\sqrt{3} + \sqrt{7 + 4\sqrt{3}} \]

### Conclusion

The final answer is:

\[ x^2 + x - 9 = 4\sqrt{3} - 2 + \sqrt{7 + 4\sqrt{3}} \]