SOLUTION: Solve for z: z^2-2z+1=-4

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Question 139800: Solve for z: z^2-2z+1=-4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

z%5E2-2z%2B1=-4 Start with the given equation


z%5E2-2z%2B1%2B4=0 Add 4 to both sides.


z%5E2-2z%2B5=0 Combine like terms


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


Starting with the general quadratic

az%5E2%2Bbz%2Bc=0

the general solution using the quadratic equation is:

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



So lets solve z%5E2-2%2Az%2B5=0 ( notice a=1, b=-2, and c=5)




z+=+%28--2+%2B-+sqrt%28+%28-2%29%5E2-4%2A1%2A5+%29%29%2F%282%2A1%29 Plug in a=1, b=-2, and c=5



z+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A1%2A5+%29%29%2F%282%2A1%29 Negate -2 to get 2



z+=+%282+%2B-+sqrt%28+4-4%2A1%2A5+%29%29%2F%282%2A1%29 Square -2 to get 4 (note: remember when you square -2, you must square the negative as well. This is because %28-2%29%5E2=-2%2A-2=4.)



z+=+%282+%2B-+sqrt%28+4%2B-20+%29%29%2F%282%2A1%29 Multiply -4%2A5%2A1 to get -20



z+=+%282+%2B-+sqrt%28+-16+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



z+=+%282+%2B-+4%2Ai%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



z+=+%282+%2B-+4%2Ai%29%2F%282%29 Multiply 2 and 1 to get 2



After simplifying, the quadratic has roots of

z=1+%2B+2%2Ai or z=1+-+2%2Ai