Question 979693


You are given an equation


{{{6x^2-3px+5}}} = {{{0}}}. 


Apply the &nbsp;<B><U>Viete's theorem</U></B>&nbsp; (see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/Solving-quadratic-equations-without-quadratic-formula.lesson>Solving quadratic equations without quadratic formula</A>&nbsp; in this site):


According to this theorem, 

&nbsp;&nbsp;&nbsp;&nbsp;a) &nbsp;the sum of the roots of the quadratic equation is equal to the coefficient at &nbsp;<B>x</B>&nbsp; taken with the opposite sign and divided by the coefficient at {{{x^2}}}:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x[1]}}} + {{{x[2]}}} = {{{3p/6}}} = {{{p/2}}}. 


&nbsp;&nbsp;&nbsp;&nbsp;b) &nbsp;the sum of the roots is equal to the constant term divided by the coefficient at {{{x^2}}}:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x[1]}}}.{{{x[2]}}} = {{{5/6}}}.


Now, &nbsp;the condition of the problem requires that 


{{{x[1]}}} + {{{x[2]}}} = {{{x[1]}}}.{{{x[2]}}},


which implies that 


{{{p/2}}} = {{{5/6}}}. 


Hence, &nbsp;p = {{{2*(5/6)}}} = {{{5/3}}}.


As a last step, &nbsp;substitute this value of &nbsp;<B>p</B>&nbsp; into the original equation, &nbsp;find its roots and make sure that the roots satisfy to the imposed condition. 

Please, &nbsp;&nbsp;<B>do it yourself</B>. &nbsp;&nbsp;&nbsp;&nbsp;Good luck!