If the sum of the roots of the quadratic equation `ax^(2)+bx+c=0` is equal to the sum of the squares of their reciprocals,then prove that `2a^(2)c=c^(2)b+b^(2)a`

Explanation

Let the roots of the quadratic equation `ax^(2)+bx+c=0` be `alpha` and `beta`.
Then `alpha+beta- - b/a` and `alpha beta=c/a` ……….1
From the given condition
`alpha+beta=1/(alpha^(2))+1/(beta^(2))`………..2
`1/(alpha^(2))+1/(beta^(2))=((alpha^(2)+beta^(2)))/(alpha^(2)beta^(2))`.................3
`alpha^(2)+beta^(2)=(alpha+beta)^(2)-2alpha beta`
`=(-b/a)^(2)-(2c)/a`.......[From 1]
`=(b^(2))/(a^(2))-(2c)/a`
`=(b^(2)-2ca)/(a^(2))` ............4
`1/(alpha^(2))+1/(beta^(2))=(alpha^(2)+beta^(2))/(alpha^(2)beta^(2))=(b^(2)-2ca)/(a^(2)):-(c^(2))/(a^(2))` ..........[From 4 and 1]
`=(b^(2)-2ca)/(a^(2))xx(a^(2))/(c^(2))=(b^(2)-2ca)/(c^(2))`
`:.1/(alpha^(2))+1/(beta^(2))=(b^(2)-2ca)/(c^(2))`.............5
`alpha+beta=1/(alpha^(2))+1/(beta^(2))`
`:.-b/a=(b^(2)-2ca)/(c^(2))`..........[From 1 and 5]
`:.-bc^(2)=ab^(2)-2ca^(2)`
`:.2ca^(2)=ab^(2)+bc^(2)`
i.e `2ca^(2)=bc^(2)+ab^(2)`