Solving the Equation (x + 3)² = 64
This equation is a quadratic equation in disguise, and we can solve it using a few different methods:
Method 1: Square Root Property

Take the square root of both sides: √(x + 3)² = ±√64

Simplify: x + 3 = ±8

Solve for x:
 x + 3 = 8 => x = 5
 x + 3 = 8 => x = 11
Therefore, the solutions to the equation (x + 3)² = 64 are x = 5 and x = 11.
Method 2: Expanding and Solving

Expand the left side: x² + 6x + 9 = 64

Move all terms to one side: x² + 6x  55 = 0

Factor the quadratic: (x + 11)(x  5) = 0

Set each factor to zero and solve:
 x + 11 = 0 => x = 11
 x  5 = 0 => x = 5
Again, we arrive at the solutions x = 5 and x = 11.
Verifying the Solutions
We can plug our solutions back into the original equation to verify they are correct:
 For x = 5: (5 + 3)² = 8² = 64 (True)
 For x = 11: (11 + 3)² = (8)² = 64 (True)
Both solutions hold true for the original equation.
In summary, the solutions to the equation (x + 3)² = 64 are x = 5 and x = 11.