Question:
What is the sum of all the roots of the equation $x|x|-5|x+2|+6=0$?
My Method of Solving:
3 cases arise here.
Case 1:
Considering that $x\geq0$, the equation becomes $x^2-5(x+2)+6=x^2-5x-4=0$
Using the fact that sum of roots = $-\frac{b}{a}$, I get $\mathbf{5}$. Let's say this is $\mathbf{S_1=5}$. This is for the 1st set of roots.
Case 2:
Considering that $-2\leq x\leq0$, the equation becomes $-x^2-5x-4=0=x^2+5x+4\space(\times-1)$
Again, using the same fact, sum of roots = $\mathbf{-5}$. Let's say this is $\mathbf{S_2=-5}$. This is for the 2nd set of roots.
Case 3:
Considering that $x\leq -2$, the equation becomes $-x^2+5x+16=0$.
Again, sum of roots = $\mathbf{5}$, so $\mathbf{S_3=5}$, this is the 3rd set of roots.
So sum of all the roots = $S_1+S_2+S_3=5$.
Am I right?
No it is not correct as in the first case you have taken $x\ge 0 $ but one of the two roots is negative in the first case so we will take only 1 value. Similarly for other two cases only one value comes out to be valid for each case. So find those values and so the sum accordingly.