If x and y are the possible solutions to
x+sqrt(y) = sqrt(17)
y+sqrt(x) = sqrt(17)
find (x+y)
Let
x+sqrt(y) = sqrt(17)
y+sqrt(x) = sqrt(17)
As x and y are not stated to be different, and given the symmetry, let x=y
Then
sqrt(x)(sqrt(x) + 1) = sqrt(17)
(x+sqrt(x))^2=17
Using the quadratic formula:
Let p=sqrt(x)
p^2+p-sqrt(17)=0; when a=1, b=1, c=sqrt(17)
p=-(1)+/-sqrt(1+4*1*sqrt(17))/2
p=1/2(-1+sqrt(1+4sqrt(17))), is the positive solution.
So x = p^2 = 1/4(-1+sqrt(1+4sqrt(17)))^2, around 2.532
Checking: 1/4(-1+sqrt(1+4sqrt(17)))^2+1/2(-1+sqrt(1+4sqrt(17))) =
1/4 (-1 + sqrt(1 + 4 sqrt(17))) (1 + sqrt(1 + 4 sqrt(17))) =
sqrt(17), as expected.
Then (x+y)=1/2(-1+sqrt(1+4sqrt(17)))^2, around 5.0638
Edited on October 18, 2023, 8:10 am
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Posted by broll
on 2023-10-18 08:06:57 |
Possible Solution
