SOLUTION: For the following, solve. Find exact solutions {{{y^4-3y^2=-1}}}

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Question 142982: For the following, solve. Find exact solutions

y%5E4-3y%5E2=-1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
y%5E4-3y%5E2=-1 Start with the given equation


y%5E4-3y%5E2%2B1=0 Add 1 to both sides


Let x=y%5E2. So this means that x%5E2=y%5E4


x%5E2-3x%2B1=0 Replace y%5E4 with x%5E2. Replace y%5E2 with x.


Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve x%5E2-3%2Ax%2B1=0 ( notice a=1, b=-3, and c=1)




x+=+%28--3+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29 Plug in a=1, b=-3, and c=1



x+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29 Negate -3 to get 3



x+=+%283+%2B-+sqrt%28+9-4%2A1%2A1+%29%29%2F%282%2A1%29 Square -3 to get 9 (note: remember when you square -3, you must square the negative as well. This is because %28-3%29%5E2=-3%2A-3=9.)



x+=+%283+%2B-+sqrt%28+9%2B-4+%29%29%2F%282%2A1%29 Multiply -4%2A1%2A1 to get -4



x+=+%283+%2B-+sqrt%28+5+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%283+%2B-+sqrt%285%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%283+%2B-+sqrt%285%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%283+%2B+sqrt%285%29%29%2F2 or x+=+%283+-+sqrt%285%29%29%2F2



So the solutions in terms of x are



x+=+%283+%2B+sqrt%285%29%29%2F2 or x+=+%283+-+sqrt%285%29%29%2F2



But remember, we let x=y%5E2. So this means


y%5E2+=+%283+%2B+sqrt%285%29%29%2F2 or y%5E2+=+%283+-+sqrt%285%29%29%2F2


Take the square root of both sides in each case

y+=0%2B-sqrt%28+%283+%2B+sqrt%285%29%29%2F2%29 or y+=0%2B-sqrt%28+%283+-+sqrt%285%29%29%2F2%29


So after breaking up the expressions, we get the four solutions

y+=+sqrt%28+%283+%2B+sqrt%285%29%29%2F2%29, y+=+-sqrt%28+%283+%2B+sqrt%285%29%29%2F2%29, y+=sqrt%28+%283+-+sqrt%285%29%29%2F2%29, y+=-sqrt%28+%283+-+sqrt%285%29%29%2F2%29