Question

The roots of the equation
$(3-x{)}^4+(2-x{)}^4$ = $(5-2x{)}^4$ are

• A. all real
• B. all imaginary
• C. two real and two imaginary
• D. none of these

Answer 1

Answer: (C)

$(3-x{)}^4+(2-x{)}^4=(5-2x{)}^4=(3-x+2-x{)}^4$
$\Rightarrow (3-x{)}^4+(2-x{)}^4=(3-x{)}^4+(2-x{)}^4$
$+4(3-x{)}^3(2-x)+6(3-x{)}^2(2-x{)}^2$
$+4(3-x)(2-x{)}^3$
$\Rightarrow 2(3-x{)}^3(2-x)+3(3-x{)}^2(2-x{)}^2$
$+2(3-x)(2-x{)}^3=0$
$\Rightarrow x=2,3$
or $2(3-x{)}^2+3(3-x)(2-x)+2(2-x{)}^2=0$
$\Rightarrow 2x^2-35x+44=0$
Since its disc, is $<0$.
$\therefore$ it has no real roots.
Hence the given = $n$ has two real roots and two imaginary roots.