Conditions for an expression to be a perfect square

Question

Find the relation in $a,c$ and $d$ so that

$$(x^4 + ax^3 + bx^2 + cx + d)$$

is perfect square where $a,b,c,d \in \mathbb{R}$

Answer 1

Hmm... never thought of this before. So let's dig in and see what happens.

But $(x^2+mx+n)^2=$

$x^4+2mx^3 + (2n+m^2)x^2+2mnx+n^2$

So to be a perfect square $a,b,c,d $ must satisfy

$a=2m; $

$b=2n+m^2$ or $b=\pm 2\sqrt {d}+\frac {a^2}4$

$c=2mn $ or $c=\pm a\sqrt {d} $

$d=n^2$ or $d\ge 0$

Answer 2

If

$$(x^2+px+q)^2=x^4+ax^3+bx^2+cx+d$$

Then \begin{align*} 2p&=a\\ p^2+2q&=b\\ 2pq&=c\\ q^2&=d \end{align*} Hence we get two equalities: \begin{align*} a\sqrt{d}=c\quad\;\;\text{and}&\quad\;\;\left(\frac a2\right)^2+2\sqrt{d}=b\\ &\text{or}\\ -a\sqrt{d}=c\quad\;\;\text{and}&\quad\;\;\left(\frac a2\right)^2-2\sqrt{d}=b \end{align*}