The problem asks to determine the value of $X_1 \cdot X_2$, where $X_1$ and $X_2$ are the roots of the quadratic equation:

$2x^2 + x - 10 = 0$

To solve this, we can use Vieta's formulas, which relate the coefficients of a quadratic equation to the sum and product of its roots. For a general quadratic equation $ax^2 + bx + c = 0$, the formulas are:

• Sum of the roots: $X_1 + X_2 = -\frac{b}{a}$
• Product of the roots: $X_1 \cdot X_2 = \frac{c}{a}$

Here, $a = 2$, $b = 1$, and $c = -10$. Using the formula for the product of the roots:

$X_1 \cdot X_2 = \frac{c}{a} = \frac{-10}{2} = -5$

Thus, the value of $X_1 \cdot X_2$ is $-5$.

Would you like further details or explanations?

Here are five related questions to explore further:

1. How do we calculate the sum of the roots $X_1 + X_2$?
2. How can we solve this quadratic equation by factoring or using the quadratic formula?
3. What is Vieta’s theorem, and how does it apply to higher-degree polynomials?
4. How would the solution change if the equation were $2x^2 + x + 10 = 0$?
5. Can the quadratic equation $2x^2 + x - 10 = 0$ have complex roots, and under what conditions?

Tip: Vieta's formulas are a powerful tool to quickly find relationships between the roots of polynomial equations without solving for the exact roots.