If the expression `x^(2)-(m+2)x+4` is always positive for all real values of x, then find range of m

Options

A. `mlt2`
B. `mgt-6`
C. `-6ltmlt2`
D. `mlt-6ormgt`

Answer

To solve the problem, we need to determine the range of \( m \) such that the quadratic expression \( x^2 - (m+2)x + 4 \) is always positive for all real values of \( x \).

### Step-by-Step Solution:

1. **Identify the Quadratic Coefficients**:
The given quadratic expression is:
\[ x^2 - (m+2)x + 4 \]
Here, we can identify:
- \( a = 1 \)
- \( b = -(m + 2) \)
- \( c = 4 \)

2. **Condition for the Quadratic to be Always Positive**:
A quadratic expression \( ax^2 + bx + c \) is always positive if its discriminant is less than zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Therefore, we need:
\[ D < 0 \]

3. **Calculate the Discriminant**:
Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = [-(m + 2)]^2 - 4 \cdot 1 \cdot 4 \]
This simplifies to:
\[ D = (m + 2)^2 - 16 \]

4. **Set Up the Inequality**:
We need to solve the inequality:
\[ (m + 2)^2 - 16 < 0 \]
Rearranging gives:
\[ (m + 2)^2 < 16 \]

5. **Solve the Inequality**:
Taking the square root of both sides, we have:
\[ -4 < m + 2 < 4 \]
Now, subtract 2 from all parts of the inequality:
\[ -6 < m < 2 \]

6. **Conclusion**:
The range of \( m \) such that the expression \( x^2 - (m+2)x + 4 \) is always positive for all real values of \( x \) is:
\[ m \in (-6, 2) \]