What is the sum of all values of \(b\) for which \(\sqrt{(b-3)^2} = 5\)?
["# What Is the Sum of All Values of (b) for Which (\sqrt{(b−3)^2} = 5)?", "Understanding equations involving square roots and absolute values can be simplified when you recognize key mathematical principles. One common question students and math enthusiasts encounter is: What is the sum of all values of (b) for which (\sqrt{(b−3)^2} = 5)?", "### Breaking Down the Equation", "The expression (\sqrt{(b−3)^2}) represents the square root of a squared term, which simplifies directly to the absolute value of (b - 3):", "[\n\sqrt{(b−3)^2} = |b - 3|\n]", "So, the original equation becomes:", "[\n|b - 3| = 5\n]", "### Solving the Absolute Value Equation", "An absolute value equation like (|b - 3| = 5) means that the distance between (b) and 3 is 5 units. This gives two distinct solutions:", "1. (b - 3 = 5 \Rightarrow b = 8)\n2. (b - 3 = -5 \Rightarrow b = -2)", "### Finding the Sum of All Values", "The two values of (b) that satisfy the equation are (8) and (-2). Adding them together:", "[\n8 + (-2) = 6\n]", "### Conclusion", "The sum of all values of (b) for which (\sqrt{(b−3)^2} = 5) is:", "[\n\boxed{6}\n]", "This problem highlights how absolute value equations hidden inside square roots can be resolved by translating them into distance-based comparisons, making the solution both elegant and intuitive. Whether you're preparing for exams or mastering algebra, understanding these foundational steps enhances problem-solving skills."]
