The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is_________
The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is_________
Solution
Case I : $x \geq-5$
$\begin{aligned}
& x^2+5 x+2 x+12=0 \\
& x^2+7 x+12=0 \\
& x=-3,-4
\end{aligned}$
Case II : $-7 < \mathrm{x} < -5$
$-x^2-5 x+2 x+14-2=0$
$\begin{aligned}
& -x^2-3 x+12=0 \\
& x=\frac{-3 \pm \sqrt{9+48}}{2} \\
& =\frac{-3 \pm \sqrt{57}}{2} \\
& x=\frac{-3-\sqrt{57}}{2}, \frac{-3+\sqrt{57}}{2} \text { (rejected) }
\end{aligned}$
Case III : $\mathrm{x} \leq-7$
$\begin{aligned}
& -x^2-5 x-2 x-14-2=0 \\
& x^2+7 x+16=0 \\
& D=49-64 < 0
\end{aligned}$
No solutions
No. of solutions $=3$
Asked in: JEE Main 2024 (05 Apr Shift 2)
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