<s>[INST] Are there any two real nonzero numbers $a$ and $b$ such that $a^2+ab+b^2 = 0$ Why or why not? Are there any two real nonzero numbers $a$ and $b$ such that $a^2+ab+b^2 = 0$ Why or why not?
$$a^2+ab+b^2=(a+b)^2-ab=0$$ iff $$(a+b)^2=ab \tag{1}$$
but $(a+b)^2 = a^2+2ab+b^2 $ so equation 1 couldn't possibly be true.
Also, when $a=b\ne 0$, $(a^2+ab+b^2)(a-b) = a^3-b^3 =0$.
 [/INST] One way is to start from $$a^2+ab+b^2=b^2\left(\left(\frac{a}{b}\right)^2+\frac{a}{b}+1\right)=0$$ so letting $x=\frac{a}{b}$ we get the quadratic equation $x^2+x+1=0$ which has no real roots. This is similar to the idea hinted at in the comment by RiverX15.
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