If both the roots of the quadratic equation `x^(2) - mx + 4 = 0` are real and distinct and they lie in the interval [1, 5], then m lies in the interval

Options

A. (4, 5]
B. (3, 4)
C. (5, 6)
D. `(-5, -4)`

Answer

To solve the problem, we need to find the values of \( m \) such that both roots of the quadratic equation \( x^2 - mx + 4 = 0 \) are real, distinct, and lie within the interval \([1, 5]\).

### Step 1: Ensure the roots are real and distinct
For the quadratic equation \( ax^2 + bx + c = 0 \), the roots are real and distinct if the discriminant \( D \) is greater than zero. The discriminant is given by:

\[ D = b^2 - 4ac \]

In our case, \( a = 1 \), \( b = -m \), and \( c = 4 \). Therefore, the discriminant becomes:

\[ D = (-m)^2 - 4 \cdot 1 \cdot 4 = m^2 - 16 \]

We require:

\[ m^2 - 16 > 0 \]

### Step 2: Solve the inequality
The inequality \( m^2 - 16 > 0 \) can be factored as:

\[ (m - 4)(m + 4) > 0 \]

To solve this inequality, we find the critical points where \( m - 4 = 0 \) and \( m + 4 = 0 \), which gives us \( m = 4 \) and \( m = -4 \). We can test the intervals:

1. \( m < -4 \)
2. \( -4 < m < 4 \)
3. \( m > 4 \)

Testing these intervals:

- For \( m < -4 \) (e.g., \( m = -5 \)): \((m - 4)(m + 4) = (-9)(-1) > 0\) (True)
- For \( -4 < m < 4 \) (e.g., \( m = 0 \)): \((m - 4)(m + 4) = (-4)(4) < 0\) (False)
- For \( m > 4 \) (e.g., \( m = 5 \)): \((m - 4)(m + 4) = (1)(9) > 0\) (True)

Thus, the solution to the inequality is:

\[ m < -4 \quad \text{or} \quad m > 4 \]

### Step 3: Ensure the roots lie in the interval [1, 5]
Next, we need to ensure that both roots lie in the interval \([1, 5]\). Let the roots be \( r_1 \) and \( r_2 \). By Vieta's formulas, we know:

- The sum of the roots \( r_1 + r_2 = m \)
- The product of the roots \( r_1 r_2 = 4 \)

Since both roots must be in the interval \([1, 5]\), we have:

1. \( r_1 + r_2 = m \) implies \( 2 \leq m \leq 10 \) (since \( r_1, r_2 \) must be at least 1 and at most 5).
2. The product \( r_1 r_2 = 4 \) implies \( r_1, r_2 \) must be positive.

### Step 4: Combine the conditions
From the conditions derived:

1. From the discriminant condition, we have \( m < -4 \) or \( m > 4 \).
2. From the roots lying in the interval, we have \( 2 \leq m \leq 10 \).

Combining these, we find:

- The only overlapping interval is \( 4 < m \leq 10 \).

### Final Answer
Thus, the values of \( m \) that satisfy both conditions are:

\[ m \in (4, 10] \]