Question

The condition for which $a^2{x}^4+b{x}^3+c{x}^2+dx+f^2$ may be a perfect square, is

• A. $4a^{2}c-b^2=8a^{3}f$
• B. $4a^{2}c=8a^{3}f$
• C. $2a^{3}c=a^{3}f$
• D. none of these

Answer 1

Answer: (A)

We have, $a^2{x}^4+b{x}^3+c{x}^2+dx+f^2$
$={(a{x}^2+cx+f)}^{2}a$ a perfect square $=a^2{x}^4+2ac{x}^3+(2af+c^2)x^2+2cfx+f^2$
$\therefore$ $b=2ac,c=2af+c^2,d=2cf$
and Again $4a^{2}c=4a^2(2af+c^2)=8a^{3}f+b^2$
$(\because b=2ac)$
$\therefore$ $4a^{2}c=b^2+8a^{3}f$