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4-2.Quadratic Equations and Inequations
hard
સમીકરણ $x|x+5|+2|x+7|-2=0$ ના વાસ્તવિક ઉકેલોની સંખ્યા ............ છે.
A
$9$
B
$3$
C
$6$
D
$1$
(JEE MAIN-2024)
Solution
Case $I$ : $x \geq-5$
$ x^2+5 x+2 x+12=0 $
$ x^2+7 x+12=0 $
$ x=-3,-4$
Case $II$ : $-7<\mathrm{x}<-5$
$-x^2-5 x+2 x+14-2=0$
$ -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) }$
Case $III$ : $\mathrm{x} \leq-7$
$ -x^2-5 x-2 x-14-2=0 $
$ x^2+7 x+16=0 $
$ D=49-64<0$
No solutions
No. of solutions $=3$
Standard 11
Mathematics
