Q2 Let $a$ be a real number. For the two quadratic functions in $x$
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + a ^ { 2 } - a , \\ & g ( x ) = 4 - x ^ { 2 } , \end{aligned}$$
answer the following questions.
(1) The range of the values of $a$ such that the equation $f ( x ) = g ( x )$ has two different solutions is
$$- \mathbf { K } < a < \mathbf { L } .$$
(2) In the case of (1), the parabolas $y = f ( x )$ and $y = g ( x )$ intersect at two points. We are to find the range of the values of $a$ such that both of the $y$ coordinates of these points of intersection are positive.
First, let $h ( x ) = f ( x ) - g ( x )$. Since the solutions of the equation $f ( x ) = g ( x )$ are the $x$ coordinates of the points of intersection of parabolas $y = f ( x )$ and $y = g ( x )$, the solutions of $h ( x ) = 0$ have to be between $- \mathbf { M }$ and $\mathbf { N }$. Accordingly, we have
$$\begin{aligned} & h \left( - \mathbf { M } \right) = a ^ { 2 } - \mathbf { O } a + \mathbf { P } > 0 , \quad \cdots \cdots \cdots (1)\\ & h ( \mathbf { N } ) = a ^ { 2 } + \mathbf { Q } a + \mathbf { R } > 0 . \quad \cdots \cdots \cdots (2) \end{aligned}$$
Also, from the position of the axis of the parabola $y = h ( x )$ we have that
$$- \mathbf { S } < a < \mathbf { T } . \quad \cdots \cdots \cdots (3)$$
Therefore, from (1), (2), (3) and (4) we obtain
$$- \mathbf { U } < a < \mathbf{V}.$$