To solve the quadratic equation $x^2 - 7x - 8 = 0$, we can use the quadratic formula. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

For your equation $x^2 - 7x - 8 = 0$, the values of $a$, $b$, and $c$ are:

$a = 1, \quad b = -7, \quad c = -8$

The quadratic formula is:

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

Now, let's plug in the values of $a$, $b$, and $c$:

$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-8)}}{2(1)}$

Simplifying the terms:

$x = \frac{7 \pm \sqrt{49 + 32}}{2}$

$x = \frac{7 \pm \sqrt{81}}{2}$

$x = \frac{7 \pm 9}{2}$

Now, we have two possible solutions:

1. $x = \frac{7 + 9}{2} = \frac{16}{2} = 8$
2. $x = \frac{7 - 9}{2} = \frac{-2}{2} = -1$

Thus, the two solutions are:

$x = 8 \quad \text{or} \quad x = -1$

Would you like any further details or have any questions?

Here are five related questions for deeper understanding:

1. How would you solve this using factoring instead of the quadratic formula?
2. What does the discriminant tell you about the number of solutions to a quadratic equation?
3. Can you find the vertex of the parabola represented by $x^2 - 7x - 8$?
4. What would happen if the discriminant was negative?
5. How can we graphically represent the solutions of this quadratic equation?

Tip: Always check the discriminant $b^2 - 4ac$ to determine if the quadratic equation has real, equal, or complex solutions before solving!