Under what conditions is `2x^(2)+kx+2` always positive ?

Answer

To determine under what conditions the quadratic expression \(2x^2 + kx + 2\) is always positive, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression can be compared to the standard form \(ax^2 + bx + c\). Here, we have:
- \(a = 2\)
- \(b = k\)
- \(c = 2\)

### Step 2: Check the leading coefficient
Since \(a = 2\) is greater than 0, the parabola opens upwards. This is a necessary condition for the quadratic to be always positive.

**Hint:** A quadratic function opens upwards if the coefficient of \(x^2\) (which is \(a\)) is positive.

### Step 3: Calculate the discriminant
The discriminant \(D\) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
Substituting our values:
\[ D = k^2 - 4(2)(2) = k^2 - 16 \]

### Step 4: Set the condition for positivity
For the quadratic to be always positive, the discriminant must be less than 0:
\[ k^2 - 16 < 0 \]

### Step 5: Solve the inequality
Rearranging the inequality gives:
\[ k^2 < 16 \]
Taking the square root of both sides, we get:
\[ -4 < k < 4 \]

### Conclusion
Thus, the quadratic expression \(2x^2 + kx + 2\) is always positive when:
\[ k \in (-4, 4) \]

**Final Answer:** The quadratic \(2x^2 + kx + 2\) is always positive for \(k\) in the interval \((-4, 4)\).

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