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We are tasked with finding the set of values of \( k \) such that the quadratic inequality

\[
(k^2 + 36)x^2 + (k - 6)x + 0.5 \geq 0
\]

holds for all real values of \( x \).

### Step 1: Analyze the general form of the quadratic inequality
This inequality is of the form:

\[
ax^2 + bx + c \geq 0
\]

where:
- \( a = k^2 + 36 \)
- \( b = k - 6 \)
- \( c = 0.5 \)

For the inequality to hold for all real values of \( x \), the quadratic must either have no real roots (i.e., the discriminant must be negative) or it must be a perfect square with no sign changes (i.e., the quadratic is always non-negative).

### Step 2: Discriminant Condition
The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c \) is given by:

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

Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:

\[
\Delta = (k - 6)^2 - 4(k^2 + 36)(0.5)
\]

Simplify the expression:

\[
\Delta = (k - 6)^2 - 2(k^2 + 36)
\]

Now expand and simplify further:

\[
\Delta = (k^2 - 12k + 36) - 2k^2 - 72
\]
\[
\Delta = k^2 - 12k + 36 - 2k^2 - 72
\]
\[
\Delta = -k^2 - 12k - 36
\]

### Step 3: Find values of \( k \) where the discriminant is non-positive
For the quadratic to be non-negative for all real \( x \), we require the discriminant to be non-positive:

\[
-k^2 - 12k - 36 \leq 0
\]

Multiply through by \( -1 \) (which flips the inequality sign):

\[
k^2 + 12k + 36 \geq 0
\]

Now, factor this expression:

\[
(k + 6)^2 \geq 0
\]

This inequality is always true since the square of any real number is non-negative. Therefore, the discriminant is always non-positive.

### Step 4: Conclusion
Since the discriminant is always non-positive, the quadratic has no real roots or has a double root, and the coefficient of \( x^2 \) is always positive (\( k^2 + 36 > 0 \) for all \( k \)), the inequality holds for all real values of \( x \) for any value of \( k \).

Thus, **the inequality is valid for all values of \( k \)**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the discriminant help in determining the nature of a quadratic inequality?
2. What conditions ensure that a quadratic expression is non-negative for all real values of \( x \)?
3. Can you explain the impact of the coefficient of \( x^2 \) on the shape of a quadratic function?
4. What happens if the discriminant equals zero?
5. How would the solution change if the inequality were strict (i.e., \( > 0 \) instead of \( \geq 0 \))?

**Tip**: Always check the sign of the quadratic's leading coefficient to determine whether it opens upwards or downwards when analyzing inequalities.

Find the set of values of k for which (k^2 + 36)x^2 + (k - 6)x + 0.5 ≥ 0 for all real values of x.

Learn how to find the set of values of k such that the quadratic inequality (k^2 + 36)x^2 + (k - 6)x + 0.5 ≥ 0 holds for all real values of x. This problem involves quadratic discriminants and inequality analysis, suitable for high school students.

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