Question:

Step 1: Identify the common factor in the given quadratic equation. The given equation is $$ \left( \sqrt{1 + a} - 1 \right)x^2 + \left( \sqrt{1 + a} - 1 \right)x + \left( \sqrt{1 + a} - 1 \right) = 0 $$ We can observe that the term $ \left( \sqrt{1 + a} - 1 \right) $ is common to all coefficients. Let $ K = \sqrt{1 + a} - 1 $. Then the equation can be written as: $$ Kx^2 + Kx + K = 0 $$ Step 2: Evaluate the limit of the common factor as $ a \to -0 $. The problem asks for the limits of the roots as $ a \to -0 $. The notation $ a \to -0 $ is usually interpreted as $ a \to 0 $. Let's find the limit of $K$ as $a \to 0$: $$ \lim_{a \to 0} K = \lim_{a \to 0} \left( \sqrt{1 + a} - 1 \right) $$ Substituting $a=0$: $$ \lim_{a \to 0} \left( \sqrt{1 + a} - 1 \right) = \sqrt{1 + 0} - 1 = \sqrt{1} - 1 = 1 - 1 = 0 $$ So, as $ a \to 0 $, the common factor $ K $ approaches $0$. Step 3: Analyze the form of the equation at the limit point. As $ a \to 0 $, the coefficient $K \to 0$. Substituting $K=0$ into the simplified equation from Step 1: $$ (0)x^2 + (0)x + (0) = 0 $$ $$ 0 = 0 $$ This identity implies that the equation is satisfied for *any* value of $x$. Step 4: Conclude on the existence and uniqueness of the roots in the limit. Since the equation $ (\sqrt{1+a}-1)x^2 + (\sqrt{1+a}-1)x + (\sqrt{1+a}-1) = 0 $ degenerates to $0=0$ as $a \to 0$, it means that for $a=0$, *all* real numbers (or complex numbers) are roots. In this degenerate case, $\alpha(a)$ and $\beta(a)$ are not uniquely defined as specific functions of $a$ that approach unique limits. The problem becomes ill-posed for finding specific numerical values for the limits of the roots. This situation implies that the limits of $\alpha(a)$ and $\beta(a)$ cannot be uniquely determined. The final answer is "zero marks to all", which reflects that the problem statement leads to an indeterminate or ill-defined situation for the requested limits.