To determine how many equations of the form $px^2 + qx + 2 = 0$ have real roots where $p$ and $q$ belong to the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$, we need to ensure that the discriminant of the quadratic equation is non-negative. Recall that for a quadratic equation of the form $ax^2 + bx + c = 0$, the discriminant is given by:

$D = b^2 - 4ac$

In our case, we have $a = p$, $b = q$, and $c = 2$. Therefore, the discriminant becomes:

$D = q^2 - 4 \times p \times 2 = q^2 - 8p$

For the equation to have real roots, the discriminant must be greater than or equal to zero:

$q^2 - 8p \geq 0$

Rearranging this inequality gives:

$q^2 \geq 8p$

Now, we will check all combinations of $p$ and $q$ from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ to see when this condition holds:

1. If $p = 1$, then $q^2 \geq 8$. The possible values of $q$ are $3, 4, 5, 6, 7, 8$ (since $q^2$ must be at least 8).
2. If $p = 2$, then $q^2 \geq 16$. The possible values of $q$ are $4, 5, 6, 7, 8$.
3. If $p = 3$, then $q^2 \geq 24$. The possible values of $q$ are $5, 6, 7, 8$.
4. If $p = 4$, then $q^2 \geq 32$. The possible values of $q$ are $6, 7, 8$.
5. If $p = 5$, then $q^2 \geq 40$. The possible values of $q$ are $7, 8$.
6. If $p = 6$, then $q^2 \geq 48$. The possible values of $q$ are $7, 8$.
7. If $p = 7$, then $q^2 \geq 56$. The possible values of $q$ are $8$.
8. If $p = 8$, then $q^2 \geq 64$. The possible values of $q$ are $8$.

Next, we count all valid combinations:

• $p = 1$: 6 values of $q$ (3, 4, 5, 6, 7, 8)
• $p = 2$: 5 values of $q$ (4, 5, 6, 7, 8)
• $p = 3$: 4 values of $q$ (5, 6, 7, 8)
• $p = 4$: 3 values of $q$ (6, 7, 8)
• $p = 5$: 2 values of $q$ (7, 8)
• $p = 6$: 2 values of $q$ (7, 8)
• $p = 7$: 1 value of $q$ (8)
• $p = 8$: 1 value of $q$ (8)

Total number of combinations is:

$6 + 5 + 4 + 3 + 2 + 2 + 1 + 1 = 24$

Hence, there are 24 equations of the form $px^2 + qx + 2 = 0$ that have real roots.

Would you like to see further details or have any questions? Here are some related problems you might find interesting:

1. What if the constant term is changed to 1? How many such equations have real roots?
2. Explore the discriminant condition for different ranges of $p$ and $q$.
3. What happens if the values of $p$ and $q$ are restricted to odd numbers?
4. Consider the case when the coefficient of $x^2$ is allowed to be negative.
5. How many equations have exactly one real root?

Tip: For a quadratic equation to have real roots, always check if its discriminant is non-negative.