If `x=1/(2-sqrt(3)),` find the value of `x^3-2x^2-7x+5`

Explanation

To find the value of \( x^3 - 2x^2 - 7x + 5 \) given that \( x = \frac{1}{2 - \sqrt{3}} \), we will follow these steps:

### Step 1: Rationalize the denominator
We start with:
\[ x = \frac{1}{2 - \sqrt{3}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator \( 2 + \sqrt{3} \):
\[ x = \frac{1 \cdot (2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]

### Step 2: Simplify the denominator
Using the difference of squares:
\[ (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, we have:
\[ x = 2 + \sqrt{3} \]

### Step 3: Find \( x^2 \)
Now, we calculate \( x^2 \):
\[ x^2 = (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 4: Find \( x^3 \)
Next, we calculate \( x^3 \):
\[ x^3 = (2 + \sqrt{3})(7 + 4\sqrt{3}) = 2 \cdot 7 + 2 \cdot 4\sqrt{3} + \sqrt{3} \cdot 7 + \sqrt{3} \cdot 4\sqrt{3} \]
Calculating each term:
\[ = 14 + 8\sqrt{3} + 7\sqrt{3} + 12 = 26 + 15\sqrt{3} \]

### Step 5: Substitute into the expression
Now we substitute \( x^3 \), \( x^2 \), and \( x \) into the expression \( x^3 - 2x^2 - 7x + 5 \):
\[ x^3 - 2x^2 - 7x + 5 = (26 + 15\sqrt{3}) - 2(7 + 4\sqrt{3}) - 7(2 + \sqrt{3}) + 5 \]

### Step 6: Simplify the expression
Calculating \( -2x^2 \):
\[ -2(7 + 4\sqrt{3}) = -14 - 8\sqrt{3} \]
Calculating \( -7x \):
\[ -7(2 + \sqrt{3}) = -14 - 7\sqrt{3} \]
Now substituting:
\[ = (26 + 15\sqrt{3}) - (14 + 8\sqrt{3}) - (14 + 7\sqrt{3}) + 5 \]
Combining like terms:
\[ = 26 - 14 - 14 + 5 + (15\sqrt{3} - 8\sqrt{3} - 7\sqrt{3}) = 3 + 0\sqrt{3} = 3 \]

### Final Answer
The value of \( x^3 - 2x^2 - 7x + 5 \) is:
\[ \boxed{3} \]