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

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

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

lioness99a
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Anshuman Kumar
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2 Answers2

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*}

Ángel Mario Gallegos
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    Don't forget q could be negative. So c could equal $-a\sqrt {d} $ and, if so b = the difference, not the sum. – fleablood Apr 10 '17 at 07:28
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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 $d\ge 0$

fleablood
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  • Could you please tell that how do you write everything in this mathematical format? I'm new here. – Anshuman Kumar Apr 10 '17 at 07:51
  • @Anshuman we use LaTeX. – The Dead Legend Apr 10 '17 at 13:32
  • When I want to write something formatted, say $\pm\sqrt{d}$ I research what that LaTex code is. Shortcut : I can rightclick on someone else's math code and select "Show math as > TeX command". It shows me the cod for $c=\pm a\sqrt {d}$ is "c=\pm a\sqrt {d}" So I type my code between dollar signs as "$c=\pm a\sqrt {d}$" Some conventions become clear. code for symbols start with backslash, other wise it's just letters. {}will not render but will allow as to "block" more than a single letter into a command.e.g.$10^{251}$ is $10^{251}$ but $10^251$ only does the first letter $10^251$ – fleablood Apr 10 '17 at 16:28