Given the law of composition x*y=xy+4x+4y+12, solve the following equation: x*x*x*x
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calendarEducator since 2011
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Since you completed the right side of the equation a bit later, the equation that is solved is x*x*x*x = 0, hence, solving the equation x*x*x*x = 12 yields:
`(x+4)^4 - 4 = 12 => (x+4)^4 = 16 => x+4 = root(4)(16) => x+4 =+-2`
`x + 4 = 2 => x = -2`
`x + 4 = -2 => x = -6`
Hence, evaluating the solution to the equation x*x*x*x = 12 yields x = -2 and x = -6.
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calendarEducator since 2011
write5,348 answers
starTop subjects are Math, Science, and Business
Notice that you may write the law of composition such that:
`x*y = (x+4)(y+4)-4`
Resoning by analogy yields:
`x*x = (x+4)(x+4)-4 => x*x = (x+4)^2 - 4`
Using `x*x = (x+4)^2 - 4` , you may evaluate `(x*x)*(x*x)` such that:
`(x*x)*(x*x)= ((x+4)^2 - 4 + 4)((x+4)^2 - 4 + 4) - 4`
`(x*x)*(x*x) = ((x+4)^2)((x+4)^2) - 4 => (x*x)*(x*x) = (x+4)^4 - 4`
You need to solve for x the equation `(x*x)*(x*x) = 0` such that:
`(x+4)^4 - 4 = 0`
You should convert the difference of squares into a product such that:
`((x+4)^2 - 2)((x+4)^2 + 2) = 0`
You should solve the following equations such that:
`(x+4)^2 - 2 = 0 => (x+4)^2= 2 => x+4 = +-sqrt2 => x_(1,2) = -4+-sqrt2`
`(x+4)^2+ 2 = 0 => (x+4)^2 = -2 => x_(3,4) = -4+-i*sqrt2`
Hence, evaluating the solutions to the given equation yields `x_(1,2) = -4+-sqrt2` and `x_(3,4) = -4+-i*sqrt2` .
x*x*x*x = 12
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