Number of distinct real roots of equation |mathrm{x}||mathrm{x}+2|-5|mathrm{x}+1|-1=0 is............

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4-2.Quadratic Equations and Inequations

normal

The number of distinct real roots of the equation $|\mathrm{x}||\mathrm{x}+2|-5|\mathrm{x}+1|-1=0$ is....................

$3$

$9$

$4$

$6$

(JEE MAIN-2024)

Solution

Case $-1$

$ x \geq 0 $

$ x^2+2 x-5 x-5-1=0 $

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

$ x=\frac{3 \pm \sqrt{9+24}}{2}=\frac{3 \pm \sqrt{33}}{2}$

One positive root

Case $-2$

$ -1 \leq x<0 $

$ -x^2-2 x-5 x-5-1=0 $

$ x^2+7 x+6=0 $

$ (x+6)(x+1)=0 $

$ x=-1$

one root in range

Case $-3$

$ -2 \leq x<-1 $

$ x^2-2 x+5 x+5-1=0 $

$ x^2-3 x-4=0 $

$ (x-4)(x+1)=0$

No root in range

Case $-4$

$ x<-2 $

$ x^2+7 x+4=0 $

$ x=\frac{-7 \pm \sqrt{49-16}}{2}=\frac{7 \pm \sqrt{33}}{2}$

one root in range

Total number of distinct roots are $3$

Standard 11

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The number of distinct real roots of the equation $|\mathrm{x}||\mathrm{x}+2|-5|\mathrm{x}+1|-1=0$ is....................

Solution

Similar Questions

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