Conditions for an expression to be a perfect square

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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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Anshuman Kumar
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Updated on December 15, 2022

Comments

  • Anshuman Kumar
    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 $a,b,c,d \in \mathbb{R}$

    • The Dead Legend
      The Dead Legend over 6 years
      this should have roots $\alpha,\alpha,\beta,\beta$ Try applying properties of a polynomial coefficients after that. hope it helps.
    • Anshuman Kumar
      Anshuman Kumar over 6 years
      I tried doing that but couldn't solve those equations to get the condition.
    • The Dead Legend
      The Dead Legend over 6 years
      i got a^2=-2c and 2b^2=-cd. Good enough?
    • Anshuman Kumar
      Anshuman Kumar over 6 years
      The answer given in my book is c^2 = a^2d
  • fleablood
    fleablood over 6 years
    Don't forget q could be negative. So c could equal $-a\sqrt {d} $ and, if so b = the difference, not the sum.
  • Anshuman Kumar
    Anshuman Kumar over 6 years
    Could you please tell that how do you write everything in this mathematical format? I'm new here.
  • The Dead Legend
    The Dead Legend over 6 years
    @Anshuman we use LaTeX.
  • fleablood
    fleablood over 6 years
    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 $10^251$