The number of real solutions of the equation x|x+5|+2|x+7|-2=0 is __________.

Question

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is \_\_\_\_\_\_\_\_\_\_.

Answer: 3

Accepted Answer

$x|x+5|+2|x+7|-2=0$

$\begin{gathered} dx\ge -5\Rightarrow x(x+5)+2(x+7)-2=0 \ {x}^{2}+7x+12=0\Rightarrow x=-3,-4 \end{gathered}$

$\begin{gathered} \, x\in (-7,-5) \ x(-x-5)+2(x+7)-2=0 \ -{x}^{2}-3x+12=0 \ \Rightarrow {x}^{2}+3x-12=0 \ \Rightarrow x=\frac{-3-\sqrt{57}}{2} $

$ \, x\le -7 \ \Rightarrow x(-x-5)+2(-x-7)-2=0 \ -{x}^{2}-7x-16=0\Rightarrow {x}^{2}+7x+16=0 \end{gathered}$

No solution

Solutions · 1