The given equation is $(C - 7)^2 = 81$.

To solve for $C$, follow these steps:

Step 1: Take the square root of both sides

Take the square root of both sides of the equation to eliminate the square:

$\sqrt{(C - 7)^2} = \pm \sqrt{81}$

This simplifies to:

$C - 7 = \pm 9$

Step 2: Solve for $C$

Now, solve the equation for $C$ by considering the two possible cases:

1. $C - 7 = 9$
2. $C - 7 = -9$

Case 1: $C - 7 = 9$

Add 7 to both sides:

$C = 9 + 7 = 16$

Case 2: $C - 7 = -9$

Add 7 to both sides:

$C = -9 + 7 = -2$

Final Answer

The solutions for $C$ are:

$C = 16 \quad \text{or} \quad C = -2$

Would you like more details or have any questions?

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Here are some related questions you might be interested in:

1. How do you solve equations involving absolute values, like $|C - 7| = 9$?
2. What is the process for solving quadratic equations like $(C - 3)^2 = 49$?
3. How do you solve equations where both sides are squared, such as $(2C + 1)^2 = (C - 3)^2$?
4. What steps would you take to solve $(C + 5)^2 = 36$?
5. How do you solve quadratic equations using the method of completing the square?
6. What is the significance of the ± symbol in solutions involving square roots?
7. How do you solve quadratic equations using the quadratic formula?
8. What happens if the equation involves a perfect square trinomial?

Tip: Always remember to consider both the positive and negative square roots when solving equations like $(x - a)^2 = b$.