If `alpha, beta` are the roots of the equation `x^(2)-7x+1=0` then value of
`{(1)/((alpha-7)^(2))+(1)/(beta-7)^(2))} - {((alpha^(2)+1)/(beta^(2)+1))+((beta^(2)+1)/(alpha^(2)+1))}` is

Options

A. `0`
B. `1`
C. `47`
D. `17`

Explanation

To solve the problem, we need to evaluate the expression given the roots \( \alpha \) and \( \beta \) of the quadratic equation \( x^2 - 7x + 1 = 0 \).

### Step 1: Find the roots \( \alpha \) and \( \beta \)
Using the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -7 \), and \( c = 1 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 1 \cdot 1 = 49 - 4 = 45 \]
Thus, the roots are:
\[ \alpha, \beta = \frac{7 \pm \sqrt{45}}{2} = \frac{7 \pm 3\sqrt{5}}{2} \]

### Step 2: Calculate \( \frac{1}{(\alpha - 7)^2} + \frac{1}{(\beta - 7)^2} \)
First, we find \( \alpha - 7 \) and \( \beta - 7 \):
\[ \alpha - 7 = \frac{7 + 3\sqrt{5}}{2} - 7 = \frac{-7 + 3\sqrt{5}}{2} \]
\[ \beta - 7 = \frac{7 - 3\sqrt{5}}{2} - 7 = \frac{-7 - 3\sqrt{5}}{2} \]

Now, we compute \( \frac{1}{(\alpha - 7)^2} \) and \( \frac{1}{(\beta - 7)^2} \):
\[ (\alpha - 7)^2 = \left(\frac{-7 + 3\sqrt{5}}{2}\right)^2 = \frac{(-7 + 3\sqrt{5})^2}{4} = \frac{49 - 42\sqrt{5} + 45}{4} = \frac{94 - 42\sqrt{5}}{4} \]
\[ (\beta - 7)^2 = \left(\frac{-7 - 3\sqrt{5}}{2}\right)^2 = \frac{(-7 - 3\sqrt{5})^2}{4} = \frac{49 + 42\sqrt{5} + 45}{4} = \frac{94 + 42\sqrt{5}}{4} \]

Thus,
\[ \frac{1}{(\alpha - 7)^2} = \frac{4}{94 - 42\sqrt{5}}, \quad \frac{1}{(\beta - 7)^2} = \frac{4}{94 + 42\sqrt{5}} \]

Adding these fractions:
\[ \frac{1}{(\alpha - 7)^2} + \frac{1}{(\beta - 7)^2} = \frac{4(94 + 42\sqrt{5}) + 4(94 - 42\sqrt{5})}{(94 - 42\sqrt{5})(94 + 42\sqrt{5})} \]
The numerator simplifies to:
\[ 4(188) = 752 \]
The denominator simplifies to:
\[ (94)^2 - (42\sqrt{5})^2 = 8836 - 8820 = 16 \]
Thus,
\[ \frac{1}{(\alpha - 7)^2} + \frac{1}{(\beta - 7)^2} = \frac{752}{16} = 47 \]

### Step 3: Calculate \( \frac{\alpha^2 + 1}{\beta^2 + 1} + \frac{\beta^2 + 1}{\alpha^2 + 1} \)
Using the equations derived from the quadratic:
\[ \alpha^2 = 7\alpha - 1 \quad \text{and} \quad \beta^2 = 7\beta - 1 \]
Thus,
\[ \alpha^2 + 1 = 7\alpha \quad \text{and} \quad \beta^2 + 1 = 7\beta \]
Now substituting:
\[ \frac{7\alpha}{7\beta} + \frac{7\beta}{7\alpha} = \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} \]
Using \( \alpha + \beta = 7 \) and \( \alpha \beta = 1 \):
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 49 - 2 = 47 \]
Thus,
\[ \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{47}{1} = 47 \]

### Step 4: Combine results
Now we can combine the results:
\[ \frac{1}{(\alpha - 7)^2} + \frac{1}{(\beta - 7)^2} - \left( \frac{\alpha^2 + 1}{\beta^2 + 1} + \frac{\beta^2 + 1}{\alpha^2 + 1} \right) = 47 - 47 = 0 \]

### Final Answer
The value of the expression is \( \boxed{0} \).