Question: What is the sum of all values of $b$ for which $\sqrt{(b-3)^2} = 5$?
["Title: Solving $\sqrt{(b-3)^2} = 5$: Find All Values of $b$ and Their Sum", "Solving equations involving square roots can sometimes be tricky, especially when the square root expression appears with a square. In this article, we walk through a clear and insightful solution to the equation $\sqrt{(b-3)^2} = 5$, explaining each step while emphasizing key mathematical concepts — and most importantly, how to find the sum of all valid solutions for $b$.", "---", "### What Does $\sqrt{(b-3)^2}$ Mean?", "The expression $\sqrt{(b-3)^2}$ simplifies to the absolute value of $b - 3$:", "$$\n\sqrt{(b-3)^2} = |b - 3|\n$$", "So, the equation becomes:", "$$\n|b - 3| = 5\n$$", "---", "### Solving the Absolute Value Equation", "The equation $|b - 3| = 5$ tells us that the distance between $b$ and 3 is exactly 5. This gives two cases:", "Case 1:\n$$\nb - 3 = 5 \quad \Rightarrow \quad b = 8\n$$", "Case 2:\n$$\nb - 3 = -5 \quad \Rightarrow \quad b = -2\n$$", "Thus, the two solutions are $b = 8$ and $b = -2$.", "---", "### Finding the Sum of All Values of $b$", "Now, we compute the sum of all valid solutions:", "$$\n8 + (-2) = 6\n$$", "So, the sum of all values of $b$ satisfying the original equation is 6.", "---", "### Why This Sum Matters", "Understanding the sum of roots in absolute value equations helps with algebraic reasoning and problem-solving. It confirms that even when solving non-linear equations, symmetry and cases play a crucial role — and checking each branch ensures no solution is missed.", "---", "### Final Answer", "$$\n\boxed{6}\n$$", "If you're studying algebra, mastering absolute value equations like this strengthens your foundation and prepares you for more complex expressions involving roots and inequalities.", "---", "Keywords: $\sqrt{(b-3)^2} = 5$, absolute value equation, sum of solutions, solve $|b - 3| = 5$, algebraic problem-solving, math tutorial, high school algebra", "Meta Description:\nDiscover how to solve $\sqrt{(b-3)^2} = 5$ by simplifying to $|b - 3| = 5$, finding all values of $b$, and calculating their sum. Learn key steps and why this matters in algebra."]
