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Solve for x: (sqrt(x) + sqrt(sqrt(x)))^2 - 10sqrt(x) - 11 = 0

User Sunshinejr
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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.

User Dln
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