Question 1177487

If {{{a^(1/2)-a^(-1/2)=1}}}, show that {{{a+a^-1=3}}}

{{{a^(1/2)-a^(-1/2)=1}}}

{{{sqrt (a)-1/a^(1/2)=1}}}

{{{sqrt (a)-1/sqrt (a)=1}}}

{{{((sqrt (a))^2-1)/sqrt (a)=1}}}

{{{(a-1)/sqrt (a)=1}}}.........square both sides

{{{(a-1)^2/(sqrt (a))^2=1^2}}}

{{{(a-1)^2/a=1}}}

{{{(a-1)^2=a}}}

{{{a^2-2a+1=a}}}

{{{a^2-2a+1-a=0}}}

{{{a^2-3a+1=0}}}

{{{a^2-3a+1=0}}}


using quadratic formula we get solutions:
 
{{{a = 3/2 + sqrt(5)/2}}} or

{{{a = 3/2 - sqrt(5)/2}}}


show that {{{a+a^-1=3}}}


{{{a+1/a=3}}}


{{{3/2 + sqrt(5)/2+1/(3/2 + sqrt(5)/2)=3}}}


{{{3/2 + sqrt(5)/2+1/((3+ sqrt(5))/2)=3}}}


{{{3/2 + sqrt(5)/2+2/(3 + sqrt(5)) =3}}}


{{{3/2 + (sqrt(5)(3 + sqrt(5))+2*2)/(2(3 + sqrt(5))) =3}}}


{{{3/2 + (3sqrt(5) + 5+4)/(2(3 + sqrt(5))) =3}}}


{{{3/2 + (9+3sqrt(5) )/(2(3 + sqrt(5))) =3}}}


{{{3/2 + 3(3+sqrt(5) )/(2(3 + sqrt(5))) =3}}}


{{{3/2 + 3cross((3+sqrt(5)) )/(2cross((3 + sqrt(5)))) =3}}}


{{{3/2 + 3/2 =3}}}


{{{6/2 =3}}}


{{{3 =3}}}