Question

The root of the equation ${\left|x\right|}^2+\left|x\right|-6=0$ are -

• A. only one real number
• B. real and sum =$1or-1$
• C. real and sum =$0$
• D. real and product =$0$

Answer 1

The correct option is C real and sum =$0$

For $x>0$

${\left|x\right|}^2+\left|x\right|-6=0$ becomes

$x^2+x-6=0$

$x^2+3x-2x-6=0$

$x(x+3)-2(x+3)=0$

$(x+3)(x-2)=0$

$\Rightarrow x=-3,2$

Roots are real and sum$=-3+2=-1$

For $x<0$

${\left|x\right|}^2+\left|x\right|-6=0$ becomes

$x^2-x-6=0$

$x^2-3x+2x-6=0$

$x(x-3)+2(x-3)=0$

$(x-3)(x+2)=0$

$\Rightarrow x=3,-2$

Roots are real and sum$=3-2=1$

Hence real and sum $=-1or1$