Question

For what value of k is the expression x^4 + kx^2 + 49 a perfect square?

Answer 1

k=± 14

Explanation:

`x^4+kx^2+49\\Disc.=b^2-4ac=k^2-4*1*49\\it ~ is ~a ~ perfect ~ square.\\k^2-196=0\\k^2=196\\k= \pm 14`

Answer 2

`k=14`

Explanation:

Completing the square gives

`x^4 + kx^2 + 49\\= (x^2+7)^2 - 14x^2 + kx^2`

So we have to cancel the latter two terms out for the expression to always become a square (otherwise we are dealing with a square plus something else, which doesn't always result in a square), giving us our answer directly.