12. Solve the equation \( (x+3)^{2}=49 \)

To solve the equation \( (x+3)^{2}=49 \), we start by taking the square root of both sides. Remember, when you take the square root, you have to consider both the positive and negative roots. This gives us: \[ x + 3 = 7 \quad \text{or} \quad x + 3 = -7 \] Solving each case, we first take \( x + 3 = 7 \): \[ x = 7 - 3 = 4 \] Then, for \( x + 3 = -7 \): \[ x = -7 - 3 = -10 \] Thus, the solutions to the equation are \( x = 4 \) and \( x = -10 \). Now, just for a little historical fun, did you know the Pythagoreans, the ancient Greek mathematicians, were among the first to study quadratic equations? They didn't have the formal methods we use today, but they knew how to manipulate shapes and values. Gardens with simple square shapes were tested based on these principles! And while you're honing your math skills, watch out for the classic mistake of forgetting the plus-minus aspect when taking square roots! Always remember – equations can sometimes hide extra solutions that are just waiting to be found. It’s like a treasure hunt for numbers!