The number of real roots of the equation x|$x - 2$| + 3|$x - 3$| +$1 = 0$is:
Step-by-Step Solution
Key Concept: Split the absolute values at$x=2$and$x=3$and solve the resulting$linear/quadratic$equations$interval-wise.$
(I)$x < 2$2$(3)$-$x + 2x - 3x + 9 + 1 = 0$2 $\Rightarrow$$x + x - 10 = 0 -1$+$\$sqrt41 -1$-$$\sqrt41$\Rightarrow x =, 2 2 \times $\sqrt (II) 2$$\le x$< 3 2 \Rightarrow $x - 2x - 3x + 9 + 1 = 0$2 \Rightarrow $x - 5x + 10 = 0$$D < 0$(III) x$\ge 3 2$x - 2x + 3x - 9 + 2 = 0$2$\Rightarrow $x + x - 8 = 0 -1$+$\$sqrt{3}2 -1$-$$\sqrt{3}2 x =, 2 2$$\times$ $\times$ 1 real roots
Correct Answer: 3