Answer:
To solve the equation (sqrt(x) + sqrt(sqrt(x)))^2 - 10sqrt(x) - 11 = 0 for x, we can use algebraic methods to simplify and isolate x.
First, let's expand the left-hand side of the equation using the distributive property:
(sqrt(x) + sqrt(sqrt(x)))^2 = (sqrt(x))^2 + 2(sqrt(x))(sqrt(sqrt(x))) + (sqrt(sqrt(x)))^2
= x + 2sqrt(x^(3/2)) + sqrt(x)
Substituting this expression back into the original equation, we get:
x + 2sqrt(x^(3/2)) + sqrt(x) - 10sqrt(x) - 11 = 0
Combining like terms, we have:
x + (sqrt(x) - 10)sqrt(x) - 11 = 0
Letting u = sqrt(x), we can rewrite this as a quadratic equation in u:
u^2 + (u-10)u - 11 = 0
Expanding and simplifying, we get:
u^2 - 9u - 11 = 0
Using the quadratic formula, we can solve for u:
u = (9 ± sqrt(9^2 + 4(11)))/2
u ≈ 10.62 or u ≈ -1.62
Since u represents a square root, it must be non-negative. Therefore, the only valid solution is:
u ≈ 10.62
Substituting back in for x, we get:
x ≈ (10.62)^2
x ≈ 112.84
Therefore, the solution to the equation (sqrt(x) + sqrt(sqrt(x)))^2 - 10sqrt(x) - 11 = 0 is approximately x = 112.84.