Question

A soma das raízes da equação |x + 2|² - |x + 2| - 2 = 0, vale:
a) -8
b) -4
c) 0
d) 4
e) 1
(Gabarito está letra D mas n entendi o porquê)

Answer 1

$\displaystyle \sf \sf |x+2|^2-|x+2|-2= 0 \\\\ Fa{\c c}amos : \\\\ |x+2| = a \\\\ Da{\'i}}: \\\\ a^2-a-2 = 0 \\\\ a^2+a-2a-2= 0 \\\\ a(a+1) - 2(a+1) = 0 \\\\ (a+1)(a-2)=0 \\\\ a+1 = \to a = - 1 \\\\ a-2 = 0 \to a = 2 \\\\ \text{desfazendo a troca de vari{\'a}vel} : \\\\ |x+2| = - 1 \ (\text{Sem solu{\c c}{\~a}o real}) \\\\ Ent{\~a}o : \\\\ |x+2| = 2 \\\\ \underline{1^\circ \text{caso. (quando a fun{\c c}{\~a}o for positivo)}} : \\\\ x+2=2 \to \boxed{\sf x = 0 }$

$\displaystyle \sf \underline{2^\circ \text{caso. (quando a fun{\c c}{\~a}o for negativa)}} : \\\\ x+2 = -2 \\\\ x=-2-2 \to \boxed{\sf x = -4}$

Soma das raízes :

$\displaystyle \sf x_1+x_2 = 0 + (-4) \\\\ \huge\boxed{\sf x_1+x_2 = -4\ }\checkmark$

letra b

(o gabarito D está errado )