Solve \(\dfrac{\sqrt{x} + \sqrt{1/x} - \sqrt{2}}{\sqrt{x} + \sqrt{1/x}} = \dfrac{\sqrt{x} + \sqrt{1/x}}{\sqrt{x} + \sqrt{1/x} + 3 \sqrt{2}}\)
+67 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)}
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
