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=(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.