The number of real roots of the equation x|x-2|+3|x-3|+1=0 is : - Vidyadip

MCQ

The number of real roots of the equation $\mathrm{x}|\mathrm{x}-2|+3|\mathrm{x}-3|+1=0$ is :

A
4
B
2
C
1
D
3

✓

Answer

C. 1
(I) $x<2$
$-x^{2}+2 x-3 x+9+1=0$
$\Rightarrow \mathrm{x}^{2}+\mathrm{x}-10=0$
$\Rightarrow \mathrm{x}=\frac{-1+\sqrt{41}}{2}, \frac{-1-\sqrt{41}}{2}$
(II) $2 \leq x<3$
$\Rightarrow \mathrm{x}^{2}-2 \mathrm{x}-3 \mathrm{x}+9+1=0$
$\Rightarrow \mathrm{x}^{2}-5 \mathrm{x}+10=0$
D $<0$
(III) $x \geq 3$
$x^{2}-2 x+3 x-9+2=0$
$\Rightarrow \mathrm{x}^{2}+\mathrm{x}-8=0$
$x=\frac{-1+\sqrt{32}}{2}, \frac{-1-\sqrt{32}}{2}$
1 real roots

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