SOLUTION: Find all real solutions of the equation. List solutions in increasing order. 2(x^-4) - 10(x^-2) + 2 = 0 Have tried to simplify and eliminate but must be missing something be

Algebra ->  Exponents -> SOLUTION: Find all real solutions of the equation. List solutions in increasing order. 2(x^-4) - 10(x^-2) + 2 = 0 Have tried to simplify and eliminate but must be missing something be      Log On


   

Question 263095: Find all real solutions of the equation. List solutions in increasing order.
2(x^-4) - 10(x^-2) + 2 = 0
Have tried to simplify and eliminate but must be missing something because I can't get anything to equal 0.
Found 2 solutions by jim_thompson5910, drk:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Hint: If we let z=x%5E%28-2%29, then z%5E2=x%5E%28-4%29. Now apply the proper substitutions to get 2z%5E2-10z%2B2=0. Now use the quadratic formula to solve 2z%5E2-10z%2B2=0 and get solutions in terms of 'z'. Once you have those solutions, plug them into z=x%5E%28-2%29 to find solutions in terms of 'x'.

Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
we are given the following equation
(i) 2%28x%5E-4%29+-+10%28x%5E-2%29+%2B+2+=+0
step 1 - multiply all terms by x^4 to get
(ii) 2+-10x%5E2+-+2x%5E4+=+0
rewrite in descending order; divide by -2 to get
(iii) x%5E4+%2B5x%5E2+-1+=+0
Let x^2 = U and we get
(iv) U%5E2+%2B5U+-1+=+0
factor using the quadratic to get
(v) U+=+%28-5+%2B-+sqrt%2825-4%2A1%2A%28-1%29%29%29%2F%282%29+
and then
(vi) U+=+%28-5%2B-+sqrt%2829%29%29%2F2
Now replace U with x^2 to get
(vii) x%5E2+=+%28-5%2B-+sqrt%2829%29%29%2F2
taking a square root gives us
(viii) x+=+%2B-sqrt%28%28-5%2B-sqrt%2829%29%29%2F2%29