The number of integral values of m for which the quadratic expression `(1 + 2m)x^(2) - 2(1 + 3m)x + 4(1 + m), x in R`, is always positive is

Options

A. 8
B. 7
C. 6
D. 3

Answer

To solve the problem, we need to find the number of integral values of \( m \) for which the quadratic expression

\[ (1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m) \]

is always positive for all \( x \in \mathbb{R} \).

### Step 1: Identify conditions for the quadratic to be always positive
A quadratic expression \( ax^2 + bx + c \) is always positive if:
1. \( a > 0 \)
2. The discriminant \( D < 0 \)

### Step 2: Determine \( a \), \( b \), and \( c \)
From the given expression, we can identify:
- \( a = 1 + 2m \)
- \( b = -2(1 + 3m) \)
- \( c = 4(1 + m) \)

### Step 3: Set the condition for \( a > 0 \)
We need \( 1 + 2m > 0 \):
\[ 2m > -1 \implies m > -\frac{1}{2} \]

### Step 4: Set the condition for the discriminant \( D < 0 \)
The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting for \( a \), \( b \), and \( c \):
\[ D = [-2(1 + 3m)]^2 - 4(1 + 2m)(4(1 + m)) \]
Calculating \( D \):
\[ D = 4(1 + 3m)^2 - 16(1 + 2m)(1 + m) \]
Expanding both terms:
\[ D = 4(1 + 6m + 9m^2) - 16(1 + 3m + 2m + 2m^2) \]
\[ D = 4 + 24m + 36m^2 - 16 - 80m - 32m^2 \]
Combining like terms:
\[ D = (36m^2 - 32m^2) + (24m - 80m) + (4 - 16) \]
\[ D = 4m^2 - 56m - 12 \]

### Step 5: Set the condition \( D < 0 \)
We need:
\[ 4m^2 - 56m - 12 < 0 \]
Dividing the entire inequality by 4:
\[ m^2 - 14m - 3 < 0 \]

### Step 6: Find the roots of the quadratic equation
Using the quadratic formula:
\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -14, c = -3 \):
\[ m = \frac{14 \pm \sqrt{196 + 12}}{2} \]
\[ m = \frac{14 \pm \sqrt{208}}{2} = \frac{14 \pm 4\sqrt{13}}{2} = 7 \pm 2\sqrt{13} \]

### Step 7: Determine the interval for \( m \)
The roots are:
\[ m_1 = 7 - 2\sqrt{13}, \quad m_2 = 7 + 2\sqrt{13} \]
The quadratic \( m^2 - 14m - 3 < 0 \) is negative between its roots:
\[ 7 - 2\sqrt{13} < m < 7 + 2\sqrt{13} \]

### Step 8: Approximate the roots
Calculating \( \sqrt{13} \approx 3.605 \):
\[ m_1 \approx 7 - 7.21 \approx -0.21, \quad m_2 \approx 7 + 7.21 \approx 14.21 \]

### Step 9: Determine integral values of \( m \)
The integral values of \( m \) in the interval \( (-0.21, 14.21) \) are:
\[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 \]
Counting these gives us \( 15 \) integral values.

### Final Answer:
The number of integral values of \( m \) for which the quadratic expression is always positive is \( \boxed{15} \).