<s>[INST] Find the value of $k$ in this question Suppose $x_1$ and $x_2$ are solutions to $x^2+x+k=0$, if $x_1^2+x_1x_2+x_2^2=2k^2$, find the value of $k$.
$x_1+x_2 = -1$
$x_1x_2 = k$ 
$(x_1+x_2)^2 = x_1^2+2(x_1x_2)+x_2^2=2k^2$ 
$=\frac {x_1^2}{2}+x_1x_2+\frac {x_2^2}{2}=2k^2=\frac{1}{2}$ 
$2k^2=\frac{1}{2}$
$k^2=\frac{1}{4}$
$k=\sqrt{\frac{1}{4}}$ 
Could someone please confirm if this is correct and perhaps identify any errors I may have made?
 [/INST] $$(x_1^2+x_2^2+x_1x_2)=(x_1+x_2)^2-x_1x_2=1-k=2k^2\implies 2k^2+k-1 \implies k=-1,1/2$$
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