SOLUTION: If a^(1/2)-a^(-1/2)=1, show that a+a^-1=3

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Question 1177487: If a^(1/2)-a^(-1/2)=1, show that a+a^-1=3
Found 3 solutions by MathLover1, ikleyn, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

If a%5E%281%2F2%29-a%5E%28-1%2F2%29=1, show that a%2Ba%5E-1=3
a%5E%281%2F2%29-a%5E%28-1%2F2%29=1
sqrt+%28a%29-1%2Fa%5E%281%2F2%29=1
sqrt+%28a%29-1%2Fsqrt+%28a%29=1
%28%28sqrt+%28a%29%29%5E2-1%29%2Fsqrt+%28a%29=1
%28a-1%29%2Fsqrt+%28a%29=1.........square both sides
%28a-1%29%5E2%2F%28sqrt+%28a%29%29%5E2=1%5E2
%28a-1%29%5E2%2Fa=1
%28a-1%29%5E2=a
a%5E2-2a%2B1=a
a%5E2-2a%2B1-a=0
a%5E2-3a%2B1=0
a%5E2-3a%2B1=0

using quadratic formula we get solutions:

a+=+3%2F2+%2B+sqrt%285%29%2F2 or
a+=+3%2F2+-+sqrt%285%29%2F2

show that a%2Ba%5E-1=3

a%2B1%2Fa=3

3%2F2+%2B+sqrt%285%29%2F2%2B1%2F%283%2F2+%2B+sqrt%285%29%2F2%29=3

3%2F2+%2B+sqrt%285%29%2F2%2B1%2F%28%283%2B+sqrt%285%29%29%2F2%29=3

3%2F2+%2B+sqrt%285%29%2F2%2B2%2F%283+%2B+sqrt%285%29%29+=3



3%2F2+%2B+%283sqrt%285%29+%2B+5%2B4%29%2F%282%283+%2B+sqrt%285%29%29%29+=3

3%2F2+%2B+%289%2B3sqrt%285%29+%29%2F%282%283+%2B+sqrt%285%29%29%29+=3

3%2F2+%2B+3%283%2Bsqrt%285%29+%29%2F%282%283+%2B+sqrt%285%29%29%29+=3

3%2F2+%2B+3cross%28%283%2Bsqrt%285%29%29+%29%2F%282cross%28%283+%2B+sqrt%285%29%29%29%29+=3

3%2F2+%2B+3%2F2+=3

6%2F2+=3

3+=3


Answer by ikleyn(52748) About Me  (Show Source):
You can put this solution on YOUR website!
.

Start from


    a^(1/2) - a^(-1/2) = 1.



Square both sides.   You will get


    a - 2 + a^(-1) = 1


    a + a^(-1) = 1 + 2 = 3,


which is what has to be proved.

At this point, the solution is completed.



Answer by MathTherapy(10549) About Me  (Show Source):
You can put this solution on YOUR website!

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