George C. answered • 11/29/12
Humboldt State and Georgetown graduate
The best substitution is u = x^(1/4). Then u^2 = x^(1/2).
The new polynomial is u^2 - u -2 =0.
(u - 2)(u + 1)=0, u = 2, -1.
x = 16, 1
Michael B.
Christy, I think George is correct, mostly. Your comment equations are strange - first you seem to be assigning the entire equation to 'u', which doesn't make much sense to me, but then you end up with the exact same equation that George did (u2 - u - 2 = 0). Either that, or you are not asking your question clearly.
HOWEVER, George, although x=1 is a result that is obtained from the solution, it is an "extraneous solution" since it doesn't solve the original equation:
11/2 - 11/4 - 2 = -2 (not 0)
The only correct real-valued answer is x=16
11/30/12
George C.
You're right, Michael. I didn't bother to check results in equation. It seemed so straight forward squaring and then squaring again that I never bothered to go back.
11/30/12
Christy W.
I understand your thinking, thats why I am having such a problem with it. I can give you an example of how the Instructor wants it. I have missed it like 5 time. This is still wrong anyway. If u = x^(1/4), then u^2 = [x^(1/4)]^2 = x^[(1/4)*(2/1)] = x^(2/4) = x^(1/2). Hence x^(1/2) - x^(1/4) - 2 = 0 is exactly the same as: u^2 - u - 2 = 0, after we replace x^(1/2) and x^(1/4) with their expressions in terms of u.11/30/12
George C.
For x, there is only one solution to this equation. Due to the degree of the equation, u^(1/4), when squared, and then squared again (raised to the 4th power) introduces an extraneous solution, namely 1. Only 16 is a solution to this equation. -1 is a solution to (u+1)(u-2)=0, but u = x^(1/4) = -1 is not equal to x. To get x you must square the value of u twice, (raise it to the 4th power). In doing so the sign changes and +1, will not solve the original equation. 16 will. Thus, there is only one solution. That is 16. x = 1 is an extraneous solution. An important concept when squaring both sides of an equation. In my haste last night I neglected this important fact.
11/30/12
Christy W.
I will let you know the correct answer.12/01/12
Christy W.
Thanks George, but that is not what I'm looking for. It will look more like this: x^1/2 - x^1/4 - 2 = 0 u = ^(1/2) u = ^(1/4) u^2 = ^(2/4) u = ^(1/2) - ^(1/4) - 2 = 0 u^2 - u - 2 = 011/29/12