If `x=(1)/(7+4sqrt(3)), y=(1)/(7-4sqrt(3))`, find the value of `5x^(2)-7xy-5y^(2)`

Answer

To solve the problem, we need to find the value of \(5x^2 - 7xy - 5y^2\) given \(x = \frac{1}{7 + 4\sqrt{3}}\) and \(y = \frac{1}{7 - 4\sqrt{3}}\).

### Step 1: Find the values of \(x\) and \(y\)

Given:
\[ x = \frac{1}{7 + 4\sqrt{3}}, \quad y = \frac{1}{7 - 4\sqrt{3}} \]

To simplify \(x\), we multiply the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{1 \cdot (7 - 4\sqrt{3})}{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} \]

Calculating the denominator:
\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1 \]
Thus,
\[ x = 7 - 4\sqrt{3} \]

Now for \(y\):
\[ y = \frac{1}{7 - 4\sqrt{3}} \Rightarrow y = \frac{1 \cdot (7 + 4\sqrt{3})}{(7 - 4\sqrt{3})(7 + 4\sqrt{3})} \]
Calculating the denominator again:
\[ (7 - 4\sqrt{3})(7 + 4\sqrt{3}) = 1 \]
Thus,
\[ y = 7 + 4\sqrt{3} \]

### Step 2: Substitute \(x\) and \(y\) into the expression \(5x^2 - 7xy - 5y^2\)

Now we have:
\[ x = 7 - 4\sqrt{3}, \quad y = 7 + 4\sqrt{3} \]

Calculating \(x^2\) and \(y^2\):
\[ x^2 = (7 - 4\sqrt{3})^2 = 49 - 56\sqrt{3} + 48 = 97 - 56\sqrt{3} \]
\[ y^2 = (7 + 4\sqrt{3})^2 = 49 + 56\sqrt{3} + 48 = 97 + 56\sqrt{3} \]

Calculating \(xy\):
\[ xy = (7 - 4\sqrt{3})(7 + 4\sqrt{3}) = 49 - 48 = 1 \]

### Step 3: Substitute into the expression

Now substituting into the expression:
\[ 5x^2 = 5(97 - 56\sqrt{3}) = 485 - 280\sqrt{3} \]
\[ -7xy = -7 \cdot 1 = -7 \]
\[ -5y^2 = -5(97 + 56\sqrt{3}) = -485 - 280\sqrt{3} \]

Combining these:
\[ 5x^2 - 7xy - 5y^2 = (485 - 280\sqrt{3}) - 7 - (485 + 280\sqrt{3}) \]
\[ = 485 - 7 - 485 - 280\sqrt{3} - 280\sqrt{3} \]
\[ = -7 - 560\sqrt{3} \]

### Final Result
Thus, the value of \(5x^2 - 7xy - 5y^2\) is:
\[ \boxed{-560\sqrt{3} - 7} \]