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 \( m \) for which both roots of the quadratic equation \( x^2 - mx + 4 = 0 \) are real, distinct, and lie within the interval \([1, 5]\).

### Step 1: Find the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For our equation \( x^2 - mx + 4 = 0 \), we have:
- \( a = 1 \)
- \( b = -m \)
- \( c = 4 \)

Thus, the discriminant is:
\[ D = (-m)^2 - 4 \cdot 1 \cdot 4 = m^2 - 16 \]

### Step 2: Ensure the roots are real and distinct
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ m^2 - 16 > 0 \]
This can be factored as:
\[ (m - 4)(m + 4) > 0 \]
To find the intervals where this inequality holds, we analyze the critical points \( m = -4 \) and \( m = 4 \).

### Step 3: Test intervals
We test the intervals determined by the critical points:
1. \( m < -4 \) (choose \( m = -5 \)): \((-5 - 4)(-5 + 4) = (-9)(-1) > 0\) (True)
2. \( -4 < m < 4 \) (choose \( m = 0 \)): \((0 - 4)(0 + 4) = (-4)(4) < 0\) (False)
3. \( m > 4 \) (choose \( m = 5 \)): \((5 - 4)(5 + 4) = (1)(9) > 0\) (True)

Thus, the solution for \( m \) from the discriminant condition is:
\[ m \in (-\infty, -4) \cup (4, \infty) \]

### Step 4: Ensure the roots lie in the interval [1, 5]
Next, we need to ensure that both roots lie in the interval \([1, 5]\). We will evaluate the quadratic at the endpoints of this interval.

1. **Evaluate at \( x = 1 \)**:
\[ f(1) = 1^2 - m \cdot 1 + 4 = 5 - m \]
For the root to be at least 1:
\[ 5 - m \geq 0 \implies m \leq 5 \]

2. **Evaluate at \( x = 5 \)**:
\[ f(5) = 5^2 - m \cdot 5 + 4 = 25 - 5m + 4 = 29 - 5m \]
For the root to be at most 5:
\[ 29 - 5m \geq 0 \implies 5m \leq 29 \implies m \leq \frac{29}{5} \]

### Step 5: Combine conditions
Now we combine the conditions:
- From the discriminant: \( m \in (-\infty, -4) \cup (4, \infty) \)
- From the roots lying in \([1, 5]\): \( m \leq 5 \) and \( m \leq \frac{29}{5} \)

Since \( \frac{29}{5} = 5.8 \), we can conclude:
\[ m \in (4, 5] \]

### Final Answer
Thus, the values of \( m \) that satisfy all conditions are:
\[ m \in (4, 5] \]