Question

p(x),f(x),g(x) are parabolas with positive coefficient, each pair of parabolas got common rational root, how to proof that p(x)+f(x)+g(x) have a rational root?

Answer 1

`{ \tt{p(x) : {y}^(2) = 4ax}} \\ { \tt{f(x) : {(y')}^(2) = 4a {x}'}} \\ { \tt{g(x) : {(y'')}^(2) = 4ax''}} \\ { \bf{since \: they've \: a \: common \: root}} : \\ {y}^( 2) = {(y')}^(2) = {(y'')}^(2) \\ = > { \tt{4ax + 4ax' + 4ax''}} \\ = 4a(x + x' + x'') \\ common \: root \: is \: 4a \\ { \tt{ \infin {}^( - ) \leqslant 4a \leqslant \infin {}^( + ) }}`