SOLUTION: x^-4-3x^-2-4=0

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Question 186689: x^-4-3x^-2-4=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E%28-4%29-3x%5E%28-2%29-4=0 Start with the given equation.


Let z=x%5E%28-2%29. So this means that z%5E2=x%5E%28-4%29


z%5E2-3z-4=0 Replace x%5E%28-4%29 with z%5E2. Replace x%5E%28-2%29 with z




Notice we have a quadratic equation in the form of az%5E2%2Bbz%2Bc where a=1, b=-3, and c=-4


Let's use the quadratic formula to solve for z


z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


z+=+%28-%28-3%29+%2B-+sqrt%28+%28-3%29%5E2-4%281%29%28-4%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-3, and c=-4


z+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%281%29%28-4%29+%29%29%2F%282%281%29%29 Negate -3 to get 3.


z+=+%283+%2B-+sqrt%28+9-4%281%29%28-4%29+%29%29%2F%282%281%29%29 Square -3 to get 9.


z+=+%283+%2B-+sqrt%28+9--16+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-4%29 to get -16


z+=+%283+%2B-+sqrt%28+9%2B16+%29%29%2F%282%281%29%29 Rewrite sqrt%289--16%29 as sqrt%289%2B16%29


z+=+%283+%2B-+sqrt%28+25+%29%29%2F%282%281%29%29 Add 9 to 16 to get 25


z+=+%283+%2B-+sqrt%28+25+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


z+=+%283+%2B-+5%29%2F%282%29 Take the square root of 25 to get 5.


z+=+%283+%2B+5%29%2F%282%29 or z+=+%283+-+5%29%2F%282%29 Break up the expression.


z+=+%288%29%2F%282%29 or z+=++%28-2%29%2F%282%29 Combine like terms.


z+=+4 or z+=+-1 Simplify.


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z+=+4 Start with the first solution


x%5E%28-2%29+=+4 Plug in z=x%5E%28-2%29


1%2Fx%5E2+=+4 Rewrite the left side with a positive exponent.


1=4x%5E2 Multiply both sides by x%5E2.


x%5E2=1%2F4 Divide both sides by 4 and rearrange the equation


x=sqrt%281%2F4%29 or x=-sqrt%281%2F4%29 Take the square root of both sides (don't forget the plus/minus)


x=1%2F2 or x=-1%2F2 Take the square root of 1%2F4 to get 1%2F2


So the first two solutions are x=1%2F2 or x=-1%2F2

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z+=+-1 Start with the second solution


x%5E%28-2%29+=+-1 Plug in z=x%5E%28-2%29


1%2Fx%5E2+=+-1 Rewrite the left side with a positive exponent.


1=-x%5E2 Multiply both sides by x%5E2


x%5E2=-1 Divide both sides by -1 and rearrange the equation.


x=sqrt%28-1%29 or x=-sqrt%28-1%29 Take the square root of both sides (don't forget the plus/minus)


x=i or x=-i Simplify (note: i=sqrt%28-1%29



So the next two solutions are x=i or x=-i


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Answer:

So the solutions are x=1%2F2, x=-1%2F2, x=i or x=-i