To solve the quadratic equation $x^2 - 4x - 45 = 0$ using completing the square, follow these steps:

Step 1: Move the constant term to the other side of the equation.

Start by isolating the $x^2$ and $x$-terms on one side.

$x^2 - 4x = 45$

Step 2: Complete the square on the left-hand side.

Take half of the coefficient of $x$ (which is $-4$) and square it.

$\frac{-4}{2} = -2 \quad \text{and} \quad (-2)^2 = 4$

Now, add 4 to both sides of the equation to complete the square.

$x^2 - 4x + 4 = 45 + 4$

$x^2 - 4x + 4 = 49$

Step 3: Rewrite the left-hand side as a perfect square.

The left-hand side can now be factored as:

$(x - 2)^2 = 49$

Step 4: Solve for $x$.

Now, take the square root of both sides:

$x - 2 = \pm \sqrt{49}$

$x - 2 = \pm 7$

Step 5: Solve for $x$ by adding 2 to both sides.

• For $x - 2 = 7$:

$x = 7 + 2 = 9$

• For $x - 2 = -7$:

$x = -7 + 2 = -5$

Final Solution:

The solutions to the equation $x^2 - 4x - 45 = 0$ are:

$x = 9 \quad \text{or} \quad x = -5$

Would you like further details on any part of this process? Here are some related questions:

1. How would the solution change if the coefficient of $x^2$ was negative?
2. How do you recognize when to complete the square versus using the quadratic formula?
3. What happens if the equation does not have a solution when completing the square?
4. Can completing the square be used for cubic or higher-order equations?
5. What is the importance of the discriminant in determining the nature of solutions?

Tip: When completing the square, always ensure that you add the same number to both sides to keep the equation balanced.