If `alpha, beta` are the roots of the equation `x^(2) - lm + m = 0`, then find the value of `1/(alpha^(2)) + 1/(beta^(2))` in terms of l and m.

Explanation

Given, `alpha, beta` are the roots of `x^(2) - lx + m = 0`
`rArr` Sum of the roots `= alpha + beta = (-(-l))/(1) = L " "(1)`
`rArr` Product of the roots `= alpha beta = (m)/(1) = m" "(2)`
Now,
`1/(alpha^(2)) + 1/(beta^(2)) = (alpha^(2) + beta^(2))/(alpha^(2)beta^(2))`
`= ((alpha^(2) + beta^(2))^(2) - 2(alphabeta))/((alphabeta)^(2))`
Substituting the values of `alpha + beta` and `alphabeta` in the above equation , we get ,
`(1)/(alpha^(2)) +(1)/(beta^(2)) = (l^(2)-2m)/(m^(2))`
`:.` The value of `1/(alpha^(2)) + 1/(beta^(2)) = (l^(2) - 2m)/(m^(2))`.