The number of common solution(s) for curves $| y | = (x - 1) (x - 2)$ and $x^{2} - 3 x - y^{2} + 2 = 0$ is

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Solution

We know $| y | = (x - 1) (x - 2)$ exists only when $(x - 1) (x - 2) \geq 0$ and neglecting the part where $(x - 1) (x - 2) < 0$ .
To get $| y | = (x - 1) (x - 2)$ , the graph is reflected about $x$ -axis when $(x - 1) (x - 2) > 0$

For, $x^{2} - 3 x - y^{2} + 2 = 0$
$\Rightarrow {(x - \frac{3}{2})}^{2} - y^{2} + 2 - \frac{9}{4} = 0 \Rightarrow {(x - \frac{3}{2})}^{2} - y^{2} = {(\frac{1}{2})}^{2}$
Which represents a rectangular Hyperbola having centre at $(\frac{3}{2}, 0)$
Thus, it can be plotted as shown below

Clearly, there are $6$ points of intersection and hence, number of solutions is $6$

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