Question: What is the sum of all values of $ a $ for which $ \sqrt{(a-3)^2} = 5 $?
["# Understanding the Sum of All Values of $ a $ for the Equation $ \sqrt{(a - 3)^2} = 5 $", "When solving equations involving square roots and absolute values, interpreting the expression correctly is key. The equation $ \sqrt{(a - 3)^2} = 5 $ may appear simple at first glance, but it reveals a powerful mathematical principle: the square root of a square equals the absolute value. Let’s explore this concept in depth and uncover the sum of all possible values of $ a $ that satisfy the equation.", "## The Meaning Behind $ \sqrt{(a - 3)^2} $", "The square root of a squared expression, $ \sqrt{x^2} $, is mathematically equivalent to $ |x| $, the absolute value of $ x $. Applying this rule:", "$$\n\sqrt{(a - 3)^2} = |a - 3|\n$$", "So, the original equation becomes:", "$$\n|a - 3| = 5\n$$", "This equation means that the distance between $ a $ and 3 on the number line is exactly 5 units.", "## Solving the Absolute Value Equation", "To solve $ |a - 3| = 5 $, we split it into two linear equations:", "1. $ a - 3 = 5 $\n2. $ a - 3 = -5 $", "Solving each:", "1. $ a = 5 + 3 = 8 $\n2. $ a = -5 + 3 = -2 $", "Thus, the two solutions are $ a = 8 $ and $ a = -2 $.", "## Finding the Sum of All Values of $ a $", "The problem asks for the sum of all values of $ a $ satisfying the equation. Adding the two solutions:", "$$\n8 + (-2) = 6\n$$", "Therefore, the sum is $ 6 $.", "## Why This Matters (Applications and Insights)", "Understanding equations like $ \sqrt{(a - 3)^2} = 5 $ goes beyond algebraic manipulation. Such expressions model real-world scenarios involving distances, deviations, or symmetries. Recognizing absolute values allows accurate problem modeling in physics, engineering, and statistics—where only magnitude matters, not direction.", "Moreover, identifying all solutions ensures nothing is missed, and computing their sum is essential in contexts like optimization or symmetric balance problems.", "## Final Summary", "- The equation $ \sqrt{(a - 3)^2} = 5 $ simplifies to $ |a - 3| = 5 $\n- Solving yields $ a = 8 $ and $ a = -2 $\n- The sum of these values is $ 8 + (-2) = 6 $\n- This result highlights the power of absolute values in capturing bidirectional distance", "Summary: The sum of all values of $ a $ satisfying $ \sqrt{(a - 3)^2} = 5 $ is $ \boxed{6} $."]
