<s>[INST] by completing the square find in terms of k the roots of the equation $x^2 + 2kx-7=0$ By completing the square find in terms of $k$ the roots of the equation $$x^2 + 2kx-7=0$$
prove for all real values of $k$, the roots are real
 [/INST] $$x^2+2kx-7=x^2+2\cdot x\cdot k+k^2-k^2-7=0$$
$$(x+k)^2=k^2+7$$
$$x+k=\pm\sqrt{k^2+7}$$
$$x=-k\pm\sqrt{k^2+7}$$
because $k^2+7\geq 0$ for all real $k$ all roots are real.
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