The given equation is:

$2x^2 - 8x - 5 = 0$

This is a quadratic equation. To solve it, we can either factor the equation, complete the square, or use the quadratic formula. Since factoring might not be straightforward, let's apply the quadratic formula, which is:

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

Here, $a = 2$, $b = -8$, and $c = -5$. Now, let's calculate the roots using this formula.

I'll compute it step by step.The solutions to the quadratic equation $2x^2 - 8x - 5 = 0$ are approximately:

$x_1 = 4.55 \quad \text{and} \quad x_2 = -0.55$

Would you like more details on how the solution was derived, or do you have any further questions?

Here are 5 related questions for you to explore:

1. How is the discriminant used to determine the nature of roots in a quadratic equation?
2. Can you explain the steps to complete the square for this equation?
3. How would the solutions change if the coefficient $a$ was altered?
4. What real-world scenarios can quadratic equations model?
5. How would you solve a quadratic equation if the discriminant is negative?

Tip: When solving quadratic equations, always check the discriminant $b^2 - 4ac$ first to determine if the solutions will be real or complex.