Let's solve the quadratic equation $x^2 - 4x - 21 = 0$ by completing the square.

Step-by-Step Process:

1. Move the constant term to the right side: $x^2 - 4x = 21$

2. Complete the square: Take half of the coefficient of $x$ (which is $-4$), divide it by 2, and then square it: $\left( \frac{-4}{2} \right)^2 = (-2)^2 = 4$ Add this value to both sides of the equation: $x^2 - 4x + 4 = 21 + 4$ $(x - 2)^2 = 25$

3. Solve for $x$: Take the square root of both sides: $x - 2 = \pm \sqrt{25}$ $x - 2 = \pm 5$

4. Find the two possible solutions: $x = 2 + 5 = 7 \quad \text{or} \quad x = 2 - 5 = -3$

Final solutions:

$x = 7 \quad \text{or} \quad x = -3$

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the general formula for completing the square?
2. How does completing the square help in deriving the quadratic formula?
3. Can every quadratic equation be solved by completing the square?
4. How is completing the square used in graphing quadratic functions?
5. Why does completing the square work for solving quadratic equations?

Tip: When completing the square, always ensure that the coefficient of $x^2$ is 1 before proceeding!