[x^2 + x + 1] = x+1

Solve the equation.

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If X^2 + X + 1 = X+1, you'd do the following:

Subtract 1 from each side. That leaves this: X^2 + X = X

Then, subtract X from each side. That leaves this: X^2 = 0

Solve for X by taking the square root; that leaves X= 0

We'll apply the definition of the whole part, which is

[a]<a<[a]+1

We note that

[x^2 + x + 1] belongs to Z

But x^2 + x + 1=x+1 => x+1 belongs to Z => x belongs to Z

By applying the definition of whole part:

[x^2 + x + 1]<x^2 + x + 1<[x^2 + x + 1]+1

But [x^2 + x + 1]=x+1

x+1<x^2 + x + 1<x+1+1

0<x^2<1

x^2 belongs to [0,1) crossed with Z set, so x=0

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