Question

solve sqrt(x)/(x*sqrt(3) + sqrt(2)) = 1/(2*x*sqrt(6) + 4).

Answer 1

To solve the equation, we will first rationalize the denominators.

Given equation: sqrt(x)/(x*sqrt(3) + sqrt(2)) = 1/(2*x*sqrt(6) + 4)

Rationalizing the denominators, we get:

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

1/(2*x*sqrt(6) + 4) * (2*x*sqrt(6) - 4)

Now rewrite the equation using the rationalized denominators:

sqrt(x)(x*sqrt(3) - sqrt(2))/(x*sqrt(3) + sqrt(2))*(x*sqrt(3) - sqrt(2)) = 1/(2*x*sqrt(6) + 4) * (2*x*sqrt(6) - 4)

Now simplify the equation further:

x*[(x*sqrt(3))^2 - (sqrt(2))^2] = 2*x*sqrt(6) - 4

x*(3x^2 - 2) = 2*x*sqrt(6) - 4

3x^3 - 2x = 2*x*sqrt(6) - 4

Now, isolate the terms containing square root on one side and rest of the terms on the other side.

3x^3 - 2x - 2*x*sqrt(6) = -4

3x^3 - 2x - 2*x(2*sqrt(6)) = -4

3x^3 - 2x - 4sqrt(6)x = -4

Now, we can solve this cubic equation by either factoring, using numerical methods or by using appropriate software.