SOLUTION: Please solve: x^6-64 = 0

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Question 119631This question is from textbook Trigonometry
: Please solve:
x^6-64 = 0 This question is from textbook Trigonometry

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E6-64=0 Start with the given equation


%28x%5E2-4%29%28x%5E4%2B4x%5E2%2B16%29=0 Factor the left side by using the difference of cubes formula


%28x%2B2%29%28x-2%29%28x%5E4%2B4x%5E2%2B16%29=0 Factor x%5E2-4 to get %28x%2B2%29%28x-2%29


Now set each factor equal to zero:

x%2B2=0, x-2=0 or x%5E4%2B4x%5E2%2B16=0

Now solve for x for the first two factors:

x=-2, x=2 or x%5E4%2B4x%5E2%2B16=0


So the first two roots are x=-2 or x=2

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Now let's solve x%5E4%2B4x%5E2%2B16=0


Let w=x%5E2

w%5E2%2B4w%2B16=0


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


Starting with the general quadratic

aw%5E2%2Bbw%2Bc=0

the general solution using the quadratic equation is:

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



So lets solve w%5E2%2B4%2Aw%2B16=0 ( notice a=1, b=4, and c=16)




w+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A1%2A16+%29%29%2F%282%2A1%29 Plug in a=1, b=4, and c=16



w+=+%28-4+%2B-+sqrt%28+16-4%2A1%2A16+%29%29%2F%282%2A1%29 Square 4 to get 16



w+=+%28-4+%2B-+sqrt%28+16%2B-64+%29%29%2F%282%2A1%29 Multiply -4%2A16%2A1 to get -64



w+=+%28-4+%2B-+sqrt%28+-48+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



w+=+%28-4+%2B-+4%2Ai%2Asqrt%283%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



w+=+%28-4+%2B-+4%2Ai%2Asqrt%283%29%29%2F%282%29 Multiply 2 and 1 to get 2



After simplifying, the quadratic has roots of

w=-2+%2B+2%2Asqrt%283%29i or w=-2+-+2%2Asqrt%283%29i

Remember, w=x%5E2. So x%5E2=-2+%2B+2%2Asqrt%283%29i or x%5E2=-2+-+2%2Asqrt%283%29i


Now take the square root of both sides for each case

x=0%2B-sqrt%28-2+%2B+2%2Asqrt%283%29i%29 or x=0%2B-sqrt%28-2+-+2%2Asqrt%283%29i%29

Now break up the plus/minus

x=sqrt%28-2+%2B+2%2Asqrt%283%29i%29, x=-sqrt%28-2+%2B+2%2Asqrt%283%29i%29, x=sqrt%28-2+-+2%2Asqrt%283%29i%29, or x=-sqrt%28-2+-+2%2Asqrt%283%29i%29

So the roots of x%5E6-64=0 are


x=-2, x=2, x=sqrt%28-2+%2B+2%2Asqrt%283%29i%29, x=-sqrt%28-2+%2B+2%2Asqrt%283%29i%29 or x=sqrt%28-2+-+2%2Asqrt%283%29i%29, or x=-sqrt%28-2+-+2%2Asqrt%283%29i%29