91Ó°ÊÓ

A key component in determining the nature of the roots in a quadratic equation is the discriminant. The discriminant is derived from the quadratic formula and is given as \(\Delta = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
In the discriminant:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (or a repeated root).
• If \(\Delta < 0\), the roots are complex, meaning they are not real numbers.

To solve a problem involving quadratic equations, determining the discriminant can give insight into how many real roots exist and thus guide us in further analysis, such as with inequalities for the roots. For the equation \(x^2 - 2mx + m^2 - 1 = 0\), the discriminant was calculated as \(\Delta = 4\). Since this is positive, we know the equation has two distinct real roots, which is crucial for understanding and solving inequalities.

The quadratic equation \(x^2 - 2mx + m^2 - 1 = 0\) needs its roots to be found to solve the problem efficiently. The roots can be determined using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Given that \(a = 1\), \(b = -2m\), and \(c = m^2 - 1\), substituting these values into the formula gives us the roots:\[x = \frac{2m \pm \sqrt{4}}{2} = \frac{2m \pm 2}{2}\]Which simplifies to:

• \(x_1 = m + 1\)
• \(x_2 = m - 1\)

These equations represent the two possible values \(x\) can take to satisfy the quadratic equation. Understanding the behavior of these roots, based on the inequalities they must satisfy, is essential for solving the problem at hand.

Inequalities play a pivotal role in determining the range of \(m\) for which both roots of the quadratic equation are greater than \(-2\) but less than \(4\). Each root of the equation, \(x_1 = m + 1\) and \(x_2 = m - 1\), needs to adhere to the inequalities:

• \(-2 < x_1 < 4\)
• \(-2 < x_2 < 4\)

Let's solve these inequalities separately:

• For \(x_1 = m + 1\):
  • \(-2 < m + 1\) simplifies to \(m > -3\)
  • \(m + 1 < 4\) simplifies to \(m < 3\)
  Thus, \(-3 < m < 3\).
• For \(x_2 = m - 1\):
  • \(-2 < m - 1\) simplifies to \(m > -1\)
  • \(m - 1 < 4\) simplifies to \(m < 5\)
  Thus, \(-1 < m < 5\).

The solution lies in the intersection of these intervals: \(-1 < m < 3\). Therefore, \(m\) must be within this range to ensure both roots meet the initial inequality requirements.