Find what values of a so that the expression `x^(2)-(a+2)x+4` is always positive.

Answer

To determine the values of \( a \) such that the expression \( x^2 - (a+2)x + 4 \) is always positive, we can follow these steps:

### Step 1: Identify the quadratic expression
The given expression is:
\[ f(x) = x^2 - (a+2)x + 4 \]

### Step 2: Analyze the quadratic expression
For a quadratic expression \( ax^2 + bx + c \) to be always positive, the following conditions must be satisfied:
1. The coefficient of \( x^2 \) (which is \( 1 \) in our case) must be positive.
2. The discriminant must be less than zero.

### Step 3: Check the coefficient of \( x^2 \)
Here, the coefficient of \( x^2 \) is \( 1 \), which is positive. So, we move to the next step.

### Step 4: Calculate the discriminant
The discriminant \( D \) of the quadratic equation \( ax^2 + bx + c \) is given by:
\[ D = b^2 - 4ac \]
For our expression:
- \( a = 1 \)
- \( b = -(a + 2) \)
- \( c = 4 \)

Substituting these values into the discriminant formula:
\[ D = (-(a + 2))^2 - 4 \cdot 1 \cdot 4 \]
\[ D = (a + 2)^2 - 16 \]

### Step 5: Set the discriminant less than zero
To ensure that the quadratic is always positive, we need:
\[ D < 0 \]
Thus, we have:
\[ (a + 2)^2 - 16 < 0 \]

### Step 6: Solve the inequality
Rearranging the inequality:
\[ (a + 2)^2 < 16 \]

Taking the square root of both sides:
\[ |a + 2| < 4 \]

This leads to two inequalities:
\[ -4 < a + 2 < 4 \]

### Step 7: Isolate \( a \)
Subtracting \( 2 \) from all parts of the inequality:
\[ -4 - 2 < a < 4 - 2 \]
\[ -6 < a < 2 \]

### Step 8: Write the final answer
Thus, the values of \( a \) for which the expression \( x^2 - (a+2)x + 4 \) is always positive are:
\[ a \in (-6, 2) \]