We are given the expressions $-8$, $x^2$, and $17x$ as the first three terms of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant.

For this case, let's set up the condition for the differences:

• The difference between the second term and the first term should be equal to the difference between the third term and the second term.

$x^2 - (-8) = 17x - x^2$

Now, let's simplify and solve this equation:

$x^2 + 8 = 17x - x^2$ $x^2 + 8 = 17x - x^2$

Add $x^2$ to both sides:

$2x^2 + 8 = 17x$

Rearrange the equation to set it equal to 0:

$2x^2 - 17x + 8 = 0$

Now we have a quadratic equation $2x^2 - 17x + 8 = 0$. We can solve this using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 2$, $b = -17$, and $c = 8$. Let's solve it.The solutions to the quadratic equation $2x^2 - 17x + 8 = 0$ are:

$x = 8 \quad \text{or} \quad x = 0.5$

Therefore, the values of $x$ that make $-8$, $x^2$, and $17x$ form the first three terms of an arithmetic sequence are $x = 8$ and $x = 0.5$.

Would you like further clarification or details on this solution?

Here are five related questions to explore:

1. What is the general formula for the $n$-th term of an arithmetic sequence?
2. How does the quadratic formula work, and why is it used to solve quadratic equations?
3. What would happen if the sequence was geometric instead of arithmetic?
4. Can there be more than two solutions in such problems involving sequences?
5. How do the values of $x^2$ and $17x$ change for different $x$ values?

Tip: Always verify that the found solutions make sense by substituting them back into the original context, such as checking the sequence's terms.