We start with the given equation:

$(x - 5)^2 = 49$

Step 1: Take the Square Root on Both Sides

To eliminate the square, take the square root of both sides:

$\sqrt{(x - 5)^2} = \pm\sqrt{49}$

$x - 5 = \pm 7$

Step 2: Solve for $x$

Now, solve for $x$ in both cases:

1. $x - 5 = 7$
   $x = 7 + 5 = 12$

2. $x - 5 = -7$
   $x = -7 + 5 = -2$

Final Answer:

$x = 12, -2$

These are the two real solutions.

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Would you like a detailed explanation of why we take both positive and negative roots?

Here are five related problems to try:

1. Solve $(x + 3)^2 = 36$.
2. Solve $(x - 7)^2 = 25$.
3. Solve $4(x - 2)^2 = 64$.
4. Solve $(2x - 1)^2 = 9$.
5. Solve $(x + 4)^2 - 16 = 0$.

Tip: Always remember that squaring a number removes its sign, so taking the square root requires considering both positive and negative values!