Question 1010881
If {{{a}}} and {{{b}}} are the roots of the quadratic equation {{{3x^2 -4x + 7 =0}}} , find:
1) {{{a/3b + b/3a}}}
2) {{{1/a^2 + 1/b^2}}}

first find the roots {{{a}}} and {{{b}}}:

{{{3x^2 -4x + 7 =0}}}.....use quadratic formula

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

{{{x = (-(-4) +- sqrt( (-4)^2-4*3*7 ))/(2*3) }}} 

{{{x = (4 +- sqrt( 16-84 ))/6 }}} 

{{{x = (4 +- sqrt( -68 ))/6 }}} 

{{{x = (4 +- i*sqrt( 4*17 ))/6 }}} 

{{{x = (4 +- 2i*sqrt( 17 ))/6 }}} 

{{{x = (cross(4)2 +- cross(2)i*sqrt( 17 ))/cross(6)3 }}}

{{{x = (2 +- i*sqrt( 17 ))/3 }}} 

so, 

{{{a = (2 + i*sqrt( 17 ))/3 }}}

{{{b = (2 - i*sqrt( 17 ))/3 }}}



1) 
{{{((2 + i*sqrt( 17 ))/3)/(3((2 - i*sqrt( 17 ))/3)) + ((2 - i*sqrt( 17 ))/3)/(3((2 + i*sqrt( 17 ))/3))}}}


{{{((2 + i*sqrt( 17 ))/3)/(cross(3)((2 - i*sqrt( 17 ))/cross(3))) + ((2 - i*sqrt( 17 ))/3)/(cross(3)((2 + i*sqrt( 17 ))/cross(3)))}}}


{{{((2 + i*sqrt( 17 ))/3)/(2 - i*sqrt( 17 )) + ((2 - i*sqrt( 17 ))/3)/(2 + i*sqrt( 17 ))}}}


{{{(2 + i*sqrt( 17 ))/(3(2 - i*sqrt( 17 ))) + (2 - i*sqrt( 17 ))/(3(2 + i*sqrt( 17 )))}}}


{{{((2 + i*sqrt( 17 ))(2 + i*sqrt( 17 )))/(3(2 - i*sqrt( 17 ))(2 + i*sqrt( 17 ))) + ((2 - i*sqrt( 17 ))(2 - i*sqrt( 17 )))/(3(2 + i*sqrt( 17 ))(2 - i*sqrt( 17 )))}}}


{{{((2 + i*sqrt( 17 ))^2 + (2 - i*sqrt( 17 ))^2)/(3(2 + i*sqrt( 17 ))(2 - i*sqrt( 17 )))}}}


{{{(4 + 2i*sqrt( 17 )+(-1)17 + 4 - 2i*sqrt( 17 )+(-1)17)/(3(2^2 - (i*sqrt( 17 ))^2))}}}



{{{(4 + cross(2i*sqrt( 17 ))-17 + 4 - cross(2i*sqrt( 17 ))-17)/(3(4 - (1)* 17 ))}}}


{{{(8+ -34)/(3(4 - (1)* 17 ))}}}


{{{-26/(3(4 +17 ))}}}


{{{-26/63}}}....exact


or approximately

{{{-0.4 }}}



2) 

{{{1/a^2 + 1/b^2}}}


{{{1/((2 + i*sqrt( 17 ))/3)^2 + 1/((2 - i*sqrt( 17 ))/3)^2}}}


{{{1/((2 + i*sqrt( 17 ))^2/9) + 1/((2 - i*sqrt( 17 ))^2/9)}}}


{{{9/(2 + i*sqrt( 17 ))^2 + 9/(2 - i*sqrt( 17 ))^2}}}


{{{-(18 (sqrt(17)-2) (2+sqrt(17)))/((sqrt(17)-2 i)^2 (sqrt(17)+2 i)^2)}}}
....

{{{-26/49}}} exact or approximately {{{-0.5}}}