We are given the following expression
[tex]x^4+kx^2+36[/tex]We need to find a value of k that will make the expression a perfect square.
Recall that a perfect square is given by
[tex](a+b)^2=a^2+2ab+b^2[/tex]Let us re-write the given expression into the above form
[tex]x^4+kx^2+36=(x^2)^2+k(x^2)+(6)^2[/tex]So, that means
[tex]\begin{gathered} a=x^2 \\ b=6 \end{gathered}[/tex]Then
[tex]2ab=2(x^2)(6)=12x^2[/tex]So, the value of k will be
[tex]\begin{gathered} kx^2=12x^2 \\ k=\frac{12x^2}{x^2} \\ k=12 \end{gathered}[/tex][tex](x^2)^2+12(x^2)+(6)^2=(x^2+6)^2[/tex]Therefore, for a value of k = 12, the given expression will be a perfect square.