Question 449686
Suppose the numbers are a,b with a+b = 4 and ab = 7. We wish to find


*[tex \LARGE \frac{1}{a^2} + \frac{1}{b^2}].


This is equal to

*[tex \LARGE \frac{b^2 + a^2}{a^2b^2}].


Right away, we know that since ab = 7, then a^2*b^2 = 49. We can find a^2 + b^2 by squaring a+b:


*[tex \LARGE (a+b)^2 = 4^2]

*[tex \LARGE a^2 + 2ab + b^2 = 16]

However we know that ab = 7, 2ab = 14, so


*[tex \LARGE a^2 + 14 + b^2 = 16 \Rightarrow a^2 + b^2 = 2]


We substitute into our desired expression to get

*[tex \LARGE \frac{1}{a^2} + \frac{1}{b^2} = \frac{a^2 + b^2}{a^2b^2} = \frac{2}{49}].