Question

For the equation $|x-4{|}^{\left(\frac{x^2-10x+24}{x-3}\right)}=1$, which among the following statement(s) is/are true?

• A. Sum of the real roots is $18.$
• B. Sum of the real roots is $11.$
• C. Product of the real roots is $90.$
• D. Product of the real roots is $30.$

Answer 1

Answer: (B), (D)

Product of the real roots is $30.$

For the exponent to be defined $x\ne 3$
If $a^m=1$, then possible cases are
$\left(i\right)a=1,m\in R$
$\left(ii\right)m=0,a\ne 0$
$\left(iii\right)a=-1,m\text{is even}$

$\left(i\right)a=1$
$\Rightarrow |x-4|=1\Rightarrow x=5,3$
$\because x\ne 3$
$\therefore x=5$

$\left(ii\right)m=0\Rightarrow \frac{x^2-10x+24}{x-3}=0$
$\Rightarrow x^2-10x+24=0$
$\Rightarrow x=4,6$ but for $x=4,|x-4|=0$
So, $x=6$

$\left(iii\right)a=-1\Rightarrow |x-4|=-1$, which is not possible as $|x-4|\ge 0$
$\therefore x=5,x=6$ are the solutions for the given equation.