무리방정식 실근

$f ( x ) = mx + 2$, $g ( x ) = mx$이고 $\displaystyle \sqrt{f ( x )} = g ( x )$의 실근은 $\alpha$에서 $\displaystyle \sqrt{m \alpha + 2} = m \alpha$, $( m \alpha - 2 ) ( m \alpha + 1 ) = 0$ ($m \alpha \geq 0$) $\therefore m \alpha = 2$ $\cdots \cdots$ $\text{㉠}$ $\displaystyle \sqrt{f \left( \frac{x}{2} \right)} = g ( x - \alpha )$의 실근은 $2$에서 $\displaystyle \sqrt{m + 2} = 2 m - m \alpha$, $\text{㉠}$을 대입하면 $\displaystyle \sqrt{m + 2} = 2 m - 2$, $( 4 m - 1 ) ( m - 2 ) = 0$ ($m \geq 1$) $\therefore m = 2$ $\therefore \alpha = 1$ $\therefore m + \alpha = 3$