We are given the quadratic equation:

$x^2 - 10x - 24 = 0$

To solve this equation using the quadratic formula, we need to recall that the quadratic formula is:

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

In this equation, $a$, $b$, and $c$ are the coefficients from the quadratic equation $ax^2 + bx + c = 0$. Here, $a = 1$, $b = -10$, and $c = -24$.

Step 1: Plug values into the quadratic formula

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(-24)}}{2(1)}$

This simplifies to:

$x = \frac{10 \pm \sqrt{100 + 96}}{2}$

$x = \frac{10 \pm \sqrt{196}}{2}$

Step 2: Simplify the square root

$\sqrt{196} = 14$

So, the equation becomes:

$x = \frac{10 \pm 14}{2}$

Step 3: Solve for the two possible values of $x$

1. $x = \frac{10 + 14}{2} = \frac{24}{2} = 12$
2. $x = \frac{10 - 14}{2} = \frac{-4}{2} = -2$

Final Answer:

The solution set is:

$\{ -2, 12 \}$

Would you like more details or have any questions? Here are five related questions to deepen your understanding:

1. How does the discriminant ($b^2 - 4ac$) help in determining the nature of the roots?
2. What would change in the solution if the discriminant was negative?
3. Can you solve a quadratic equation using the factoring method instead of the quadratic formula?
4. How does completing the square relate to the quadratic formula?
5. What is the geometric interpretation of the solutions of a quadratic equation?

Tip: Always check the discriminant before solving a quadratic equation. If it's negative, the equation will have complex solutions involving $i$.