For what value(s) of k is the inequation x^2-kx + 2 ≥ 0 true for all values of x?

We need to determine the value(s) of \( k \) for which the inequality \( x^2 - kx + 2 \geq 0 \) holds for all values of \( x \).

### Step 1: Analyze the Quadratic Expression
The inequality \( x^2 - kx + 2 \geq 0 \) represents a quadratic expression that should always be non-negative. This implies that the quadratic equation has no real roots, or it has a double root (which implies the quadratic expression is a perfect square).

### Step 2: Calculate the Discriminant
For a quadratic expression \( ax^2 + bx + c \geq 0 \), the discriminant \( \Delta \) is given by:
\[
\Delta = b^2 - 4ac
\]
For our quadratic expression \( x^2 - kx + 2 \), we have:
- \( a = 1 \)
- \( b = -k \)
- \( c = 2 \)

So, the discriminant \( \Delta \) is:
\[
\Delta = (-k)^2 - 4(1)(2) = k^2 - 8
\]

### Step 3: Ensure Non-Negativity of the Quadratic
For the quadratic to be non-negative for all \( x \), the discriminant must be non-positive:
\[
k^2 - 8 \leq 0
\]

### Step 4: Solve the Inequality
\[
k^2 \leq 8
\]
Taking the square root of both sides, we get:
\[
-\sqrt{8} \leq k \leq \sqrt{8}
\]
Since \( \sqrt{8} = 2\sqrt{2} \), the solution is:
\[
-\sqrt{8} \leq k \leq \sqrt{8}
\]
This can be written as:
\[
-\sqrt{8} \leq k \leq \sqrt{8}
\]

Thus, the values of \( k \) that satisfy the inequality \( x^2 - kx + 2 \geq 0 \) for all \( x \) are \( k \in [-2\sqrt{2}, 2\sqrt{2}] \).

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions:
1. What is the geometric interpretation of the discriminant of a quadratic equation?
2. How does the discriminant determine the nature of the roots of a quadratic equation?
3. How do you find the vertex of a quadratic function?
4. What happens if the discriminant is positive, zero, or negative?
5. Can the quadratic inequality \( x^2 - kx + 2 > 0 \) hold true for all values of \( x \)? What would be the condition?

**Tip:** Always check the discriminant first when analyzing quadratic inequalities—it gives you crucial information about the roots and the sign of the quadratic expression.

Learn how to determine the values of \( k \) for which the quadratic inequality \( x^2 - kx + 2 \geq 0 \) holds true for all values of \( x \). Explore the discriminant method and its application in analyzing quadratic inequalities. Suitable for advanced high school students.

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