Question

For what value of k is the expression x^4 + kx^2 + 36 a perfect square.

Answer 1

We are given the following expression

$x^4+kx^2+36$

We need to find a value of k that will make the expression a perfect square.

Recall that a perfect square is given by

$(a+b)^2=a^2+2ab+b^2$

Let us re-write the given expression into the above form

$x^4+kx^2+36=(x^2)^2+k(x^2)+(6)^2$

So, that means

$\begin{gathered} a=x^2 \\ b=6 \end{gathered}$

Then

$2ab=2(x^2)(6)=12x^2$

So, the value of k will be

$\begin{gathered} kx^2=12x^2 \\ k=\frac{12x^2}{x^2} \\ k=12 \end{gathered}$$(x^2)^2+12(x^2)+(6)^2=(x^2+6)^2$

Therefore, for a value of k = 12, the given expression will be a perfect square.