Respuesta :

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.

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