Key Concepts and Formulas

• Roots of a Quadratic Equation: The roots of the quadratic equation $ax^2 + bx + c = 0$ can be found using the quadratic formula, but in some cases, factorization is simpler.
• Interval Notation: Expressing inequalities using interval notation (e.g., $a < x < b$ is represented as $x \in (a, b)$).
• Intersection of Intervals: Finding the common region between two or more intervals. This corresponds to the "and" condition in logic.

Step-by-Step Solution

1. Identify the Quadratic Equation The given quadratic equation is: $x^2 - 2mx + m^2 - 1 = 0$

Explanation: We identify the coefficients of the quadratic equation in the standard form $ax^2 + bx + c = 0$. Here, $a = 1$, $b = -2m$, and $c = m^2 - 1$.

2. Solve for the Roots of the Equation We observe that the equation can be rewritten as: $(x - m)^2 - 1 = 0$

Explanation: Recognizing that $x^2 - 2mx + m^2$ is a perfect square allows for a simpler solution compared to using the quadratic formula.

This can be factored as a difference of squares: $(x - m - 1)(x - m + 1) = 0$

Explanation: Using the difference of squares identity, $a^2 - b^2 = (a - b)(a + b)$.

Setting each factor to zero gives the roots: $x - m - 1 = 0 \implies x = m + 1$ $x - m + 1 = 0 \implies x = m - 1$

Thus, the two roots are $x_1 = m - 1$ and $x_2 = m + 1$.

Explanation: The roots are now expressed explicitly in terms of the parameter $m$.

3. Apply the Conditions on the Roots The problem states that both roots must be greater than $-2$ and less than $4$. This gives the following conditions: $-2 < x_1 < 4$ $-2 < x_2 < 4$

Explanation: We translate the problem's constraints into mathematical inequalities.

4. Formulate and Solve Inequalities for $m$

Substitute the expressions for the roots into the inequalities:

For the first root ($x_1 = m - 1$): $-2 < m - 1 < 4$

Add $1$ to all parts of the inequality: $-2 + 1 < m < 4 + 1$ $-1 < m < 5 \quad \text{(Inequality 1)}$

Explanation: We isolate $m$ to find the range of values that satisfies the condition for the first root.

For the second root ($x_2 = m + 1$): $-2 < m + 1 < 4$

Subtract $1$ from all parts of the inequality: $-2 - 1 < m < 4 - 1$ $-3 < m < 3 \quad \text{(Inequality 2)}$

Explanation: We isolate $m$ to find the range of values that satisfies the condition for the second root.

5. Combine the Intervals for $m$ Since both roots must satisfy the given conditions, we need to find the intersection of the intervals defined by Inequality 1 and Inequality 2.

Inequality 1: $m \in (-1, 5)$ Inequality 2: $m \in (-3, 3)$

The intersection of these intervals is the region where both inequalities hold true. This is: $(-1, 5) \cap (-3, 3) = (-1, 3)$

Therefore, $-1 < m < 3$.

Explanation: The intersection represents the values of $m$ that satisfy both conditions simultaneously.

Common Mistakes & Tips:

• Factorization First: Always check if a quadratic equation can be easily factored before resorting to more complex methods like the quadratic formula or location of roots criteria.
• Intersection is Key: Remember that when a condition applies to "both" roots, you must find the intersection of the intervals for the parameter.
• Careful with Inequalities: Pay attention to the direction of the inequalities and whether they are strict ($<$ or $>$) or non-strict ($\leq$ or $\geq$).

Summary:

We began by factoring the quadratic equation to find its roots in terms of $m$. We then applied the given conditions that both roots must lie between $-2$ and $4$, resulting in two sets of inequalities for $m$. The intersection of these intervals gives the range of $m$ that satisfies both conditions simultaneously, which is $-1 < m < 3$.

The final answer is \boxed{(-1, 3)}, which corresponds to option (C).