Question 921337
 

let the roots be: 

{{{x[1] }}}and {{{x[2]}}}

 the difference of the squares of the roots is {{{95}}}
{{{(x[1])^2 -( x[2])^2=95}}}.............eq.1

their difference is {{{5 }}}

 
{{{x[1] - x[2]=5}}} ............eq.2 


solve the system:

{{{(x[1])^2 -( x[2])^2=95}}}.............eq.1
{{{x[1] - x[2]=5}}} ............eq.2 
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start with {{{x[1]-x[2]=5}}} ............eq.2 ...solve for{{{ x[1] }}}

{{{x[1] =5+x[2] }}}.............substitute in eq.1

{{{(5+x[2] )^2 -( x[2])^2=95}}}.............eq.1....solve for{{{ x[2] }}}

{{{25+10x[2] +x[2] ^2 -( x[2])^2=95}}}

{{{25+10x[2] +cross(x[2] ^2) -cross(( x[2])^2)=95}}}

{{{10x[2] =95-25}}}


{{{10x[2] =70}}}

{{{x[2] =70/10}}}

{{{highlight(x[2] =7)}}}

go to{{{ x[1] =5+x[2] }}} plug in {{{x[2] =7}}}

{{{x[1] =5+7}}}

{{{highlight(x[1] =12)}}}

roots are: {{{7}}} and {{{12}}}

use zero product rule

{{{f(x)=(x-x[1])(x-x[2])}}} ...........plug in{{{ x[1] =12}}} and {{{x[2] =7}}}

{{{f(x)=(x-12)(x-7)}}}

{{{f(x)=x^2-7x-12x+7*12}}}


{{{highlight(f(x)=x^2-19x+84)}}}


graph:


{{{ graph( 600, 600, -15, 25, -10, 100, x^2-19x+84) }}}