Solution:** Start by considering the equation \(\sqrt{(b-3)^2} = 7\). Since the square root function returns the non-negative value, this implies:

["# Solving (\sqrt{(b-3)^2} = 7): A Step-by-Step Guide to Understanding Absolute Value", "Solving equations involving square roots can sometimes feel confusing—especially when the equation includes expressions like (\sqrt{(b-3)^2}). But behind this seemingly tricky setup lies a clear and logical path rooted in the properties of absolute value. In this article, we’ll break down the equation (\sqrt{(b-3)^2} = 7) systematically, explain why absolute value is involved, and clarify how to solve for (b) with confidence. Whether you're a high school student learning algebra or reviewing foundational math concepts, understanding this equation is essential to mastering more advanced topics like functions and real-number solutions.", "## Why the Square Root of a Square BECOMES Absolute Value", "The square root function, by definition, returns only the non-negative root of a number. For example, (\sqrt{9} = 3), not (-3), because the square root symbol (\sqrt{\hspace{0.4cm} \like \pm}) denotes the principal (non-negative) square root.", "Now, consider (\sqrt{x^2}). This expression represents the non-negative value whose square equals (x^2). But real numbers don’t come in negative square roots—only non-negative ones do. Therefore:", "[\n\sqrt{x^2} = |x|\n]", "The absolute value function (|x|) formally captures this idea: it returns (x) if (x \geq 0), and (-x) if (x < 0), ensuring the output is always non-negative. This definition explains why (\sqrt{b^2} = |b|) for any real number (b).", "So, when faced with (\sqrt{(b-3)^2}), we rewrite it using absolute value:", "[\n\sqrt{(b-3)^2} = |b - 3|\n]", "Substituting into the original equation:", "[\n|b - 3| = 7\n]", "## Solving the Absolute Value Equation", "The equation (|b - 3| = 7) now reads: “The distance between (b) and 3 on the number line is 7.”", "To solve an absolute value equation of the form (|x| = c) (where (c \geq 0)), we write two cases:", "1. (b - 3 = 7)\n2. (b - 3 = -7)", "Solving both:", "1. (b - 3 = 7 \Rightarrow b = 10)\n2. (b - 3 = -7 \Rightarrow b = -4)", "These solutions reflect the two points on the number line exactly 7 units left (or right) of 3: (10) and (-4).", "## Final Answer and Verification", "All solutions are verified by plugging back into the original equation:", "- For (b = 10): (\sqrt{(10 - 3)^2} = \sqrt{49} = 7) ✅\n- For (b = -4): (\sqrt{(-4 - 3)^2} = \sqrt{49} = 7) ✅", "Thus, the complete set of solutions to (\sqrt{(b-3)^2} = 7) is:", "[\n\boxed{b = 10 \quad \ ext{or} \quad b = -4}\n]", "## Why This Concept Matters Beyond Algebra", "Understanding how square roots and absolute values interact is more than academic—it’s foundational for modeling real-world problems involving distances, errors, or magnitudes, such as in physics, engineering, or data analysis. Mastering these principles ensures you’re prepared for more complex equations and functions.", "Whether you’re simplifying expressions or solving equations, remember: the square root of a square is always non-negative, and absolute value captures this behavior. With this clarity, you can confidently tackle similar challenges and deepen your mathematical fluency."]