SOLUTION: {{ x^1/2-3x^1/4+2=0 }}

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Question 217614: {{ x^1/2-3x^1/4+2=0 }}
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+x%5E%281%2F2%29-3x%5E%281%2F4%29%2B2=0+
There are a couple of keys to a relatively simple solution to this problem:
  • Recognize that 1/2 is two times 1/4
  • Recognize that because 1/2 is 2 times 1/4 and because of the rules for exponents, x%5E%281%2F2%29+=+%28x%5E%281%2F4%29%29%5E2

Once you get that x%5E%281%2F2%29+=+%28x%5E%281%2F4%29%29%5E2, the rest of the problem is fairly simple. Without seeing this, the problem would be very difficult.

Your equation is of the form q%5E2+-+3q+%2B2+=+0 (with q+=+x%5E%281%2F4%29). And we solve this by factoring (or with the quadratic formula):
%28q-2%29%28q-1%29=0
The only way for this product (multiplication) to be zero is if one of the factors is zero. So we simply state this with equations:
q-2 = 0 or q-1 = 0
Solving each we get:
q = 2 or q = 1
Of course we're not interested in "q". So we substitute back in for q:
x%5E%281%2F4%29+=+2 or x%5E%281%2F4%29+=+1
Raising each side of each equation to the 4th power:
x = 16 or x = 1

Once you've done a few of these you will not need a substitute variable. You'll see that +x%5E%281%2F2%29-3x%5E%281%2F4%29%2B2=0+ will factor into %28x%5E%281%2F4%29-2%29%28x%5E%281%2F4%29-1%29+=+0. And once you have a product equal to zero you are pretty close to the final solution.