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 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 \( x \in \mathbb{R} \), we need to ensure that the quadratic does not have any real roots and opens upwards. This can be checked using the following conditions:

1. The coefficient of \( x^2 \) (denoted as \( a \)) must be positive.
2. The discriminant (denoted as \( D \)) must be less than zero.

### Step 1: Ensure \( a > 0 \)

The coefficient \( a \) is given by:

\[ a = 1 + 2m \]

For the quadratic to open upwards, we need:

\[ 1 + 2m > 0 \]

Solving this inequality:

\[ 2m > -1 \implies m > -\frac{1}{2} \]

### Step 2: Calculate the Discriminant

The discriminant \( D \) of the quadratic is given by:

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

where \( b = -2(1 + 3m) \) and \( c = 4(1 + m) \). Thus,

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

Calculating \( b^2 \):

\[ b^2 = 4(1 + 3m)^2 = 4(1 + 6m + 9m^2) = 4 + 24m + 36m^2 \]

Calculating \( 4ac \):

\[ 4ac = 4(1 + 2m)(4(1 + m)) = 16(1 + 2m)(1 + m) = 16(1 + 3m + 2m^2) = 16 + 48m + 32m^2 \]

Now substituting back into the discriminant:

\[ D = (4 + 24m + 36m^2) - (16 + 48m + 32m^2) \]

Simplifying this:

\[ D = 4 + 24m + 36m^2 - 16 - 48m - 32m^2 \]
\[ D = 4m^2 - 24m - 12 \]

### Step 3: Set the Discriminant Less Than Zero

We need:

\[ 4m^2 - 24m - 12 < 0 \]

Dividing through by 4:

\[ m^2 - 6m - 3 < 0 \]

### Step 4: Find the Roots of the Quadratic

Using the quadratic formula:

\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{6 \pm \sqrt{36 + 12}}{2} = \frac{6 \pm \sqrt{48}}{2} = 3 \pm \sqrt{12} = 3 \pm 2\sqrt{3} \]

### Step 5: Determine the Interval

The roots are:

\[ m_1 = 3 - 2\sqrt{3}, \quad m_2 = 3 + 2\sqrt{3} \]

The quadratic \( m^2 - 6m - 3 \) opens upwards, so it is negative between its roots:

\[ 3 - 2\sqrt{3} < m < 3 + 2\sqrt{3} \]

### Step 6: Find Integral Values of \( m \)

Calculating the approximate values of the roots:

- \( \sqrt{3} \approx 1.732 \)
- \( 2\sqrt{3} \approx 3.464 \)

Thus:

\[ m_1 \approx 3 - 3.464 \approx -0.464 \]
\[ m_2 \approx 3 + 3.464 \approx 6.464 \]

The integral values of \( m \) in the interval \( (-0.464, 6.464) \) are:

\[ 0, 1, 2, 3, 4, 5, 6 \]

### Final Count of Integral Values

The integral values of \( m \) are \( 0, 1, 2, 3, 4, 5, 6 \), which gives us a total of **7 integral values**.

### Conclusion

Thus, the number of integral values of \( m \) for which the quadratic expression is always positive is:

\[ \boxed{7} \]