<s>[INST] How to find the value of this expression Suppose I have a general quadratic equation $ ax^2 + bx + c = 0 $ 
And I have $ \alpha , \beta $ as roots, then what is the simplest way to find $ (a\alpha + b)^{-2} + (a\beta + b)^{-2} $ ?
I know the formula for finding the sum and product of roots, but that doesn't helped me in finding the value.
Any good hint is welcomed.
 [/INST] The roots of the equation $ax^2+bx+c=0$ are:
$$\alpha=\frac{-b-\sqrt{b^2-4ac}}{2a}; \beta=\frac{-b+\sqrt{b^2-4ac}}{2a}.$$
Now substitute these into the expression:
$$\frac{1}{(a\alpha+b)^2} + \frac{1}{(a\beta+b)^2}=$$
$$\frac{1}{(\frac{-b-\sqrt{b^2-4ac}}{2}+b)^2} + \frac{1}{(\frac{-b+\sqrt{b^2-4ac}}{2}+b)^2}=$$
$$\frac{4}{(b-\sqrt{b^2-4ac})^2}+\frac{4}{(b+\sqrt{b^2-4ac})^2}=$$
$$\frac{4(2b^2+2(b^2-4ac))}{(4ac)^2}=\frac{b^2-2ac}{a^2c^2}.$$
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