Question

All the values of \(\mathrm{m}\) for which both roots of the equation \(\mathrm{x}^{2}-2 \mathrm{mx}+\mathrm{m}^{2}-1=0\) are greater then \(-2\) but less than 4 lie in interval (a) \(\mathrm{m}<3\) (b) \(-1>\mathrm{m}<3\) (c) \(1<\mathrm{m}<4\) (d) \(-2<\mathrm{m}<\mathrm{o}\)

Short answer

The correct option is (d) \(-2 < m < o\), where the values of \(m\) for which both roots of the equation \(x^2 - 2mx + m^2 - 1 = 0\) are greater than \(-2\) but less than \(4\) lie in this interval.

Step-by-step solution

We know that a quadratic equation in standard form is written as \(ax^2 + bx + c = 0\). In this case, \(a = 1\), \(b = -2m\), and \(c = m^2 -1\). Then we can find the vertex of the parabola, which is given by the formula \((h, k) = \left(-\frac{b}{2a}, -\frac{\Delta}{4a}\right)\), where \(\Delta = b^2 - 4ac\). #Step 2: Calculate the vertex's x-coordinate

Now, we calculate the x-coordinate of the vertex of the parabola: \(h = -\frac{-2m}{2(1)} = m\). So, the x-coordinate of the vertex of the parabola is equal to \(m\). #Step 3: Find the range of \(m\) based on the x-coordinates of the parabola's roots

Since the parabola opens upwards, and we want both roots to be within the interval of \(-2 < x < 4\), the vertex's x-coordinate must also fall within the same range. This means that \(m\) should lie in the interval \(-2 < m < 4\). #Step 4: Check the given options and find the correct one

Given the options: (a) \(m < 3\) (b) \(-1 > m < 3\) (c) \(1 < m < 4\) (d) \(-2 < m < o\) Using the information obtained in Step 3, we see that option (d) \(-2 < m < o\) matches the range of values we found for \(m\). In this case, the "o" stands for an open endpoint, meaning there is no upper bound for our interval. Answer: The correct option is (d) \(-2 < m < o\).

Key concepts

Roots of Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means they can be written in the form \( ax^2 + bx + c = 0 \). The solutions, or roots, of these equations are the values of \( x \) that make the equation true. These roots can be real or complex numbers, depending on the value of the discriminant \( \Delta = b^2 - 4ac \).

- If \( \Delta > 0 \), the equation has two distinct real roots.
- If \( \Delta = 0 \), it has exactly one real root (also known as a repeated or double root).
- If \( \Delta < 0 \), the roots are complex and conjugate to each other.

In this exercise, we need both roots to lie between \( -2 \) and \( 4 \). Ensuring this might seem straightforward, but it involves checking the inequality conditions to ensure the roots do not exceed these boundaries.

Inequality of Roots

To determine the range of \( m \) for which the roots exist within the desired interval, we need to ensure both roots are greater than \(-2\) and less than \(4\).

Using the quadratic equation from the exercise, the roots are expressed as:
\[ x_1, x_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( b = -2m \) and the discriminant has already been considered. As the parabola opens upwards (since \( a = 1 \)), it implies that the roots can only fit within this interval if the vertex's x-coordinate—the midpoint of the roots—is also within this range.

For this specific case, the condition \(-2 < m < 4\) applies because the symmetry of the parabola about the vertex ensures both roots remain inside the given interval.

Vertex of Parabola

The vertex of a parabola in the equation \( ax^2 + bx + c = 0 \) provides significant insight into the location of the parabola's minimum or maximum point. For an equation \( y = ax^2 + bx + c \), the vertex is given as \((h, k) = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\). This is essential in understanding both the direction in which the parabola opens and the relative position of the roots.

In our exercise, the vertex's x-coordinate is \( m \), calculated from \( h = -\frac{-2m}{2(1)} = m \).

Since we need both roots to be greater than \(-2\) and less than \(4\), the vertex itself must be within this same range. Thus, it substantiates our result from exploring the inequality of roots that \( m \) lies between \(-2\) and \(4\). This ensures the parabola's symmetrical axis, flowing between \(-2\) and \(4\), fulfills the required condition for maintaining both roots exclusively within the interval.