Final answer:
The difference is that finding the square of 169 is an arithmetic operation (169²), while solving p²=169 involves finding variables (p=13 and p=-13) that make the equation true. In equations with an unknown squared, there are usually two solutions, and context determines if both are meaningful.
Step-by-step explanation:
The question asks about the difference between finding the square of 169 and solving the equation p²=169. Finding the square of 169 simply means calculating 169 multiplied by itself, which is 169². However, when we solve the equation p²=169, we are trying to find the value of p which, when squared, equals 169. We can achieve this by taking the square root of both sides of the equation, which gives us two solutions since the square root of a positive number can be both positive and negative. Therefore, the solutions are p = 13 and p = -13.
Understanding this concept is critical since equations like these often require us to find the value of a variable rather than the result of an arithmetic operation. In some cases, both solutions to a squared equation are meaningful, but in other cases, only one solution is reasonable. For instance, when discussing physical scenarios like the time to travel on a freeway on-ramp, a negative time would not be meaningful, leaving us with the positive solution as the sole reasonable answer.
Whenever an equation contains an unknown squared, think about the two potential solutions and analyze the context to determine whether one or both are reasonable. In the case of finding the side length a in a right triangle with sides b, c, and using the Pythagorean Theorem, we also encounter a squared variable. To find a, we need to 'undo' the square by taking the square root, which gives us the length of side a as a positive value, since side lengths cannot be negative.