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Chapter 3: Problem 196

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

Expert verified
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

01

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
02

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) \(-2 < m < o\).

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