Question

The sum of all roots of the roots of the equations $|x-1{|}^2-5|x-1|+6=0$ is

• A. 5
• B. 4
• C. 2
• D. 6

Answer 1

The correct option is B 4

Given equation is ${|x-1|}^2-5|x-1|+6=0$
We know that ${|x-1|}^2$ will be a positive value since it is a square.
But $|x-1|$ can take both values of $(x-1)$ or $-(x-1)=1-x$
Hence, ${|x-1|}^2-5|x-1|+6=0$ can be written as both
${(x-1)}^2-5(x-1)+6=0$ and ${(x-1)}^2-5(1-x)+6=0$
On simplifying ${(x-1)}^2-5(x-1)+6=0$ we get $x^2-7x+12=0$ whose sum of roots $=-\frac{-7}{1}=7$
And on simplifying ${(x-1)}^2-5(x-1)+6=0$ we get $x^2+3x+2=0$ whose sum of roots $=-\frac{3}{1}=-3$
Thus, sum of all 4 roots $7-3=4$