Exam

JEE Main

Year

2024

Date

5 April

Shift

Morning Shift

Subject

Mathematics

Chapter

Functions

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Functions

The number of distinct real roots of the equation...

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is_______

Step-by-step solution

Case-1 $$\begin{aligned} & x \geq 0 \\ & x^2+2 x-5 x-5-1=0 \\ & x^2-3 x-6=0 \\ & x=\frac{3 \pm \sqrt{9+24}}{2}=\frac{3 \pm \sqrt{33}}{2} \end{aligned}$$ One positive root Case-2 $$\begin{aligned} & -1 \leq x < 0 \\ & -x^2-2 x-5 x-5-1=0 \\ & x^2+7 x+6=0 \\ & (x+6)(x+1)=0 \\ & x=-1 \end{aligned}$$ one root in range Case-3 $$\begin{aligned} & -2 \leq x < -1 \\ & x^2-2 x+5 x+5-1=0 \\ & x^2-3 x-4=0 \\ & (x-4)(x+1)=0 \end{aligned}$$ No root in range Case-4 $$\begin{aligned} & x < -2 \\ & x^2+7 x+4=0 \\ & x=\frac{-7 \pm \sqrt{49-16}}{2}=\frac{7 \pm \sqrt{33}}{2} \end{aligned}$$ one root in range Total number of distinct roots are 3