SOLUTION: Please help me solve this equation: {{{(x-3)^(3/2)-15(x-3)^(-1/2)+2(x-3)^(1/2)=0}}}

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Question 26985: Please help me solve this equation: %28x-3%29%5E%283%2F2%29-15%28x-3%29%5E%28-1%2F2%29%2B2%28x-3%29%5E%281%2F2%29=0
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
OK, as it stands it looks too complicated to solve. However, i notice a repeating expression in all three terms, namely %28x-3%29%5E%281%2F2%29.

Let y = %28x-3%29%5E%281%2F2%29. So,

%28x-3%29%5E%283%2F2%29-15%28x-3%29%5E%28-1%2F2%29%2B2%28x-3%29%5E%281%2F2%29=0 can be written as which is then

+y%5E3+-+15y%5E%28-1%29+%2B+2y+=+0. Instantly, this is far simpler to look at and approach.

+y%5E3+-+%2815%2Fy%29+%2B+2y+=+0
+y%5E4+-+15+%2B+2y%5E2+=+0
+y%5E4+%2B+2y%5E2+-15+=+0
+%28y%5E2+-+3%29%28y%5E2+%2B+5%29+=+0
so %28y%5E2+-+3%29+=+0 OR %28y%5E2+%2B+5%29+=+0
y%5E2+=+3 OR y%5E2+=+-5
y+=+sqrt%283%29 OR y+=+-sqrt%283%29 OR y+=+sqrt%28-5%29 OR y+=+-sqrt%28-5%29

So, we have 4 answers:
y+=+sqrt%283%29 OR y+=+-sqrt%283%29 OR y+=+sqrt%28-5%29 OR y+=+-sqrt%28-5%29

Now, we have to find what x is, so:

+%28x-3%29%5E%281%2F2%29+=+sqrt%283%29+
--> (x-3) = 3
--> x = 6

+%28x-3%29%5E%281%2F2%29+=+-sqrt%283%29+
--> (x-3) = 3
--> x = 6 again.

+%28x-3%29%5E%281%2F2%29+=+sqrt%28-5%29+
--> (x-3) = -5
--> x = -2

+%28x-3%29%5E%281%2F2%29+=+-sqrt%28-5%29+
--> (x-3) = -5
--> x = -2

So, the 2 answers are x=-2 and x=6

I would suggest you put both values back into %28x-3%29%5E%283%2F2%29-15%28x-3%29%5E%28-1%2F2%29%2B2%28x-3%29%5E%281%2F2%29 to make sure that it does indeed equal zero.

jon.