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 $
$\Rightarrow \, 2x^2- 35\,x + 44 = 0$
Since its disc, is $< 0$.
$\therefore $ it has no real roots.
Hence the given = $n$ has two real roots and two imaginary roots.