JEE Main 2024 (Online) 5th April Morning ShiftQuestion 1
The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is $\_\_\_\_\_\_\_\_$ .
Quadratic Equation · Equations with Modulus and Greatest Integer
Answer$3$
Solution
Concept. Split at $x=-2,-1,0$, where the moduli change sign. Formulas used. (1) $|t|=t$ for $t\ge0$ and $|t|=-t$ for $t<0$Step 1 - $x\ge0$. $x(x+2)-5(x+1)-1=x^2-3x-6=0$, $x=\frac{3\pm\sqrt{33}}{2}$. Only $\frac{3+\sqrt{33}}{2}\approx4.37$ is valid. Step 2 - $-1\le x<0$. $-x(x+2)-5(x+1)-1=0$, i.e. $x^2+7x+6=0$: $x=-1$ (valid) or $-6$. Step 3 - $-2\le x<-1$. $-x(x+2)+5(x+1)-1=0$, i.e. $x^2-3x-4=0$: $x=4$ or $-1$, neither in range. Step 4 - $x<-2$. $x(x+2)+5(x+1)-1=x^2+7x+4=0$, $x=\frac{-7\pm\sqrt{33}}{2}$. Only $\frac{-7-\sqrt{33}}{2}\approx-6.37$ is valid. Step 5 - Count. Three distinct real roots. Answer: 3