If `alpha, beta` are the roots of `ax^(2) + bx + c = 0` and `c != 0` find the value of `(1)/((a alpha+b)^(2))+(1)/((a beta+b)^(2))` interms of a, b, c.

Answer

To find the value of \(\frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2}\) in terms of \(a\), \(b\), and \(c\), we start with the quadratic equation \(ax^2 + bx + c = 0\), where \(\alpha\) and \(\beta\) are the roots.

### Step 1: Use Vieta's Formulas
From Vieta's formulas, we know:
- \(\alpha + \beta = -\frac{b}{a}\)
- \(\alpha \beta = \frac{c}{a}\)

### Step 2: Express \(a\alpha + b\) and \(a\beta + b\)
We can express \(a\alpha + b\) and \(a\beta + b\) as follows:
\[ a\alpha + b = -c \quad \text{(from substituting \(\alpha\) into the quadratic equation)} \]
\[ a\beta + b = -c \quad \text{(from substituting \(\beta\) into the quadratic equation)} \]

### Step 3: Rewrite the Expression
Now we can rewrite the expression we need to evaluate:
\[ \frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2} = \frac{1}{(-c)^2} + \frac{1}{(-c)^2} = \frac{1}{c^2} + \frac{1}{c^2} = \frac{2}{c^2} \]

### Step 4: Substitute for \((a\alpha + b)^2\) and \((a\beta + b)^2\)
Now we need to find \((a\alpha + b)^2\) and \((a\beta + b)^2\):
\[ (a\alpha + b)^2 = c^2 \quad \text{and} \quad (a\beta + b)^2 = c^2 \]

### Step 5: Final Expression
Thus, the final expression becomes:
\[ \frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2} = \frac{2}{c^2} \]

### Conclusion
The value of \(\frac{1}{(a\alpha + b)^2} + \frac{1}{(a\beta + b)^2}\) in terms of \(a\), \(b\), and \(c\) is:
\[ \frac{2}{c^2} \]