Find the value of X where 2+9√x/2√3-3√3x=√3+2√2
We have the equation:
2 + 9√(x) / (2√3 - 3√3x) = √3 + 2√2.
First, let's multiply both sides by (2√3 - 3√3x) to get rid of the denominator:
(2 + 9√x) = (√3 + 2√2)(2√3 - 3√3x).
Now let's expand the right side:
2√3(√3) + 2√3(2√2) - 3√3x(√3) - 3√3x(2√2).
Simplify:
6√3 + 4√6 - 9x - 6√6x = 2 + 9√x.
Now we want to collect the terms involving x on one side and the constants on the other:
9√x - 6√6x = 2 - 6√3 - 4√6.
Factor an x out of the left side and divide both sides by the expression in parentheses:
√x = (2 - 6√3 - 4√6) / (1 - 6√6).
Now square both sides to eliminate the square root:
x = (2 - 6√3 - 4√6)^2 / (1 - 12√6 + 36 * 6).
Evaluate:
x = (4 - 24√3 + 36 * 3 - 16√6 + 48√3 + 144 * 6) / (1 - 12√6 + 216).
Simplify:
x = (112 - 24√6 + 48√3) / (217 - 12√6).
Since the equation is now of the form x = (a - b√c) / (d - e√c), we can conclude that x = 112/217 is the solution.
So the value of X is 112/217.