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Solve \(\dfrac{\sqrt{x} + \sqrt{1/x} - \sqrt{2}}{\sqrt{x} + \sqrt{1/x}} = \dfrac{\sqrt{x} + \sqrt{1/x}}{\sqrt{x} + \sqrt{1/x} + 3 \sqrt{2}}\)

 Dec 19, 2019
 #1
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If you enter the following sequence into the calculator:

 

(sqrt(x)+sqrt(1/x)-sqrt(2))/(sqrt(x)+sqrt(1/x))=(sqrt(x)+sqrt(1/x))/(sqrt(x)+sqrt(1/x)+(3*sqrt(2))

 

Then the answer is shown to be:

 

{x=0, x=(((3*sqrt(2))*sqrt(x)-2)/2)}

 Dec 19, 2019
 #2
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Solve for x:
(-sqrt(2) + sqrt(1/x) + sqrt(x))/(sqrt(1/x) + sqrt(x)) = (sqrt(1/x) + sqrt(x))/(3 sqrt(2) + sqrt(1/x) + sqrt(x))

Cross multiply:
(-sqrt(2) + sqrt(1/x) + sqrt(x)) (3 sqrt(2) + sqrt(1/x) + sqrt(x)) = (sqrt(1/x) + sqrt(x))^2

Subtract (sqrt(1/x) + sqrt(x))^2 from both sides:
(-sqrt(2) + sqrt(1/x) + sqrt(x)) (3 sqrt(2) + sqrt(1/x) + sqrt(x)) - (sqrt(1/x) + sqrt(x))^2 = 0

(-sqrt(2) + sqrt(1/x) + sqrt(x)) (3 sqrt(2) + sqrt(1/x) + sqrt(x)) - (sqrt(1/x) + sqrt(x))^2 = -6 + 2 sqrt(2) sqrt(1/x) + 2 sqrt(2) sqrt(x):
-6 + 2 sqrt(2) sqrt(1/x) + 2 sqrt(2) sqrt(x) = 0

Simplify and substitute y = sqrt(x).
-6 + 2 sqrt(2) sqrt(1/x) + 2 sqrt(2) sqrt(x) = -6 + (2 sqrt(2))/(sqrt(x)) + 2 sqrt(2) sqrt(x)
 = 2 sqrt(2) y - 6 + (2 sqrt(2))/y:
2 sqrt(2) y - 6 + (2 sqrt(2))/y = 0

Bring 2 sqrt(2) y - 6 + (2 sqrt(2))/y together using the common denominator y:
(2 (sqrt(2) y^2 - 3 y + sqrt(2)))/y = 0

Divide both sides by 2:
(sqrt(2) y^2 - 3 y + sqrt(2))/y = 0

Multiply both sides by y:
sqrt(2) y^2 - 3 y + sqrt(2) = 0

Divide both sides by sqrt(2):
y^2 - 3 2^(-1/2) y + 1 = 0

Subtract 1 from both sides:
y^2 - (3 y)/sqrt(2) = -1

Add 9/8 to both sides:
y^2 - 3 2^(-1/2) y + 9/8 = 1/8

Write the left hand side as a square:
(y - 3/(2 sqrt(2)))^2 = 1/8

Take the square root of both sides:
y - 3/(2 sqrt(2)) = 1/(2 sqrt(2)) or y - 3/(2 sqrt(2)) = -1/(2 sqrt(2))

Add 3/(2 sqrt(2)) to both sides:
y = sqrt(2) or y - 3/(2 sqrt(2)) = -1/(2 sqrt(2))

Substitute back for y = sqrt(x):
sqrt(x) = sqrt(2) or y - 3/(2 sqrt(2)) = -1/(2 sqrt(2))

Raise both sides to the power of two:
x = 2 or y - 3/(2 sqrt(2)) = -1/(2 sqrt(2))

Add 3/(2 sqrt(2)) to both sides:
x = 2 or y = 1/sqrt(2)

Substitute back for y = sqrt(x):
x = 2 or sqrt(x) = 1/sqrt(2)

Raise both sides to the power of two:

 x = 2      or       x = 1/2

 Dec 19, 2019

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