Factorizing An Absolute Value Expression?

Algebra Level 3

Let the largest root of $x^2+|5x|-6=0$ be $a$ and the smaller root of $|x^2+5x|-6=0$ be $b$ .

Find $a+b$ .

Notation : $| \cdot |$ denotes the absolute value function .

1 solution

Rishabh Jain
Jul 16, 2016

First equation:- $|x|^2+5|x|-6=0\implies (|x|+6)(|x|-1)=0\implies |x|=-6,1$

Since $|x|\ge 0$ , we get $|x|=1\implies x=\pm 1$ .

Hence $x=\pm 1$ .

Second equation:-

$\begin{cases} x^2+5x-6=0 &\textrm{if } x\in (-\infty,-5)\cup (0,\infty)\\x^2+5x+6=0 & \textrm{if }x\in [-5,0]\end{cases}$

Taking care of intervals we get $x=-6,1$ in first case while in second case we get $x=-3,-2$

Hence $x=-6,-3,-2,1$ .

The larger root in first equation is $1$ while smaller in the second equation is $-6$ . Hence $-6+1=\large\boxed{-5}$ .