Question

Find the value of a, b, c which will make each of the expression $x^4+a{x}^3+b{x}^2+cx+1$ and $x^4+2a{x}^3+2b{x}^2+2cx+1$ a perfect square.

Answer 1

Let

$x^4+a{x}^3+b{x}^2+cx+1=(x^2+mx+n{)}^2$

and $x^4+2a{x}^3+2b{x}^2+2cx+1=(x^2+px+q{)}^2$

On expanding, we get

$x^4+a{x}^3+b{x}^2+cx+1=x^4+m^2{x}^2+n^2+2m{x}^3+2n{x}^2+2mnx$

Similerly, $x^4+2a{x}^3+2b{x}^2+2cx+1=x^4+p^2{x}^2+q^2+2p{x}^3+2q{x}^2+2pqx$

$a=2m,b=m^2+2n,c=2mn$ and $n^2=1⟹n=\pm 1$

$\frac{a}{c}=\frac{1}{n}=\pm 1$

$b=\frac{a^2}{4}\pm 2\dots \dots \left(1\right)$

$2a=2p,2b=p^2+2q,2c=2pq$ and $q^2=1$

$2b=a^2\pm 2b=\frac{a^2}{2}\pm 1=\frac{a^2}{4}\pm 2$ (From equation $1$)

$\frac{a^2}{4}=1$ or $-1$

$-1$ is not possible therefore $n=q=1$

and $a^2=4⟹a=\pm 2$

$c=a=\pm 2$ and $b=\frac{a^2}{4}+2=3$

Therefore possible values of $(a,b,c)$ are $(2,2,3)$ and $(-2,-2,3)$