SOLUTION: What is the root of X^2+X+1=0? I have no idea how to do roots. Thanks again!

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Question 92972: What is the root of X^2+X+1=0?
I have no idea how to do roots. Thanks again!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's use the quadratic formula to find the roots:


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%2Bx%2B1=0 ( notice a=1, b=1, and c=1)

x+=+%28-1+%2B-+sqrt%28+%281%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29 Plug in a=1, b=1, and c=1



x+=+%28-1+%2B-+sqrt%28+1-4%2A1%2A1+%29%29%2F%282%2A1%29 Square 1 to get 1



x+=+%28-1+%2B-+sqrt%28+1%2B-4+%29%29%2F%282%2A1%29 Multiply -4%2A1%2A1 to get -4



x+=+%28-1+%2B-+sqrt%28+-3+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-1+%2B-+i%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)



x+=+%28-1+%2B-+i%2Asqrt%283%29%29%2F%282%29 Multiply 2 and 1 to get 2



After simplifying, the quadratic has roots of

x=-1%2F2+%2B+sqrt%283%29%2F2%2Ai or x=-1%2F2+-+sqrt%283%29%2F2%2Ai

Notice if we graph the quadratic y=x%5E2%2Bx%2B1, we get

+graph%28+500%2C+500%2C+-15.5%2C+14.5%2C+-14.25%2C+15.75%2C+x%5E2%2Bx%2B1%29+ graph of y=x%5E2%2Bx%2B1

And we can see that there are no real roots

To visually verify the answer, check out this page to see a visual representation of imaginary roots