Conditions for an expression to be a perfect square
3,821
Solution 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 $d\ge 0$
Solution 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*}
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Author by
Anshuman Kumar
Updated on December 15, 2022Comments
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Anshuman Kumar 11 months
Find the relation in $a,c$ and $d$ so that
$$(x^4 + ax^3 + bx^2 + cx + d)$$
is perfect square where $a,b,c,d \in \mathbb{R}$
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The Dead Legend over 6 yearsthis should have roots $\alpha,\alpha,\beta,\beta$ Try applying properties of a polynomial coefficients after that. hope it helps.
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Anshuman Kumar over 6 yearsI tried doing that but couldn't solve those equations to get the condition.
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The Dead Legend over 6 yearsi got a^2=-2c and 2b^2=-cd. Good enough?
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Anshuman Kumar over 6 yearsThe answer given in my book is c^2 = a^2d
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fleablood over 6 yearsDon't forget q could be negative. So c could equal $-a\sqrt {d} $ and, if so b = the difference, not the sum.
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Anshuman Kumar over 6 yearsCould you please tell that how do you write everything in this mathematical format? I'm new here.
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The Dead Legend over 6 years@Anshuman we use LaTeX.
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fleablood over 6 yearsWhen 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 $10^251$