First, solve the equation \(\sqrt{(b-3)^2} = 5\). The square root of a square gives the absolute value:
["Solving the Equation (\sqrt{(b-3)^2} = 5): Understanding Absolute Value in Step-by-Step Detail", "When solving equations involving square roots and squared expressions, understanding the relationship between these operations and absolute values is essential. One common equation students encounter is:", "[\n\sqrt{(b - 3)^2} = 5\n]", "This equation prevents confusion by clearly linking the square root of a squared term to the absolute value of that expression. In this article, we will walk through how to solve it step-by-step and explain why the absolute value is key in the process.", "---", "### Step 1: Recognize the Meaning of the Square Root of a Square", "The key to solving (\sqrt{(b - 3)^2}) lies in recognizing that:", "[\n\sqrt{x^2} = |x|\n]", "This identity means that taking the square root of a squared number returns the non-negative (absolute) value of that number. For example, (\sqrt{9} = 3), but (\sqrt{(3)^2} = |3| = 3); similarly, (\sqrt{(-3)^2} = |-3| = 3).", "Applying this to our equation:", "[\n\sqrt{(b - 3)^2} = |b - 3|\n]", "So the equation becomes:", "[\n|b - 3| = 5\n]", "---", "### Step 2: Solve the Absolute Value Equation", "An absolute value equation of the form (|x| = a), where (a > 0), has two solutions:", "[\nx = a \quad \ ext{or} \quad x = -a\n]", "Applying this to our equation:", "[\n|b - 3| = 5 \quad \Rightarrow \quad b - 3 = 5 \quad \ ext{or} \quad b - 3 = -5\n]", "---", "### Step 3: Solve Each Case", "Case 1: (b - 3 = 5)", "Add 3 to both sides:", "[\nb = 5 + 3 = 8\n]", "Case 2: (b - 3 = -5)", "Add 3 to both sides:", "[\nb = -5 + 3 = -2\n]", "---", "### Step 4: Write the Final Solution", "The two solutions to the original equation (\sqrt{(b - 3)^2} = 5) are:", "[\nb = 8 \quad \ ext{and} \quad b = -2\n]", "---", "### Why Absolute Value Matters", "In equations involving (\sqrt{x^2}), overlooking the absolute value can lead to incorrect solutions. Remember: square roots output only non-negative values, even when the expression inside the square is negative. The absolute value ensures the output is always correct whether the base is positive or negative.", "---", "### Summary", "- Start with (\sqrt{(b-3)^2} = 5)\n- Use the identity (\sqrt{x^2} = |x|) to rewrite as (|b - 3| = 5)\n- Solve (b - 3 = \pm 5)\n- Find (b = 8) or (b = -2)", "Understanding absolute values and square roots’ behavior helps avoid common mistakes and builds a strong foundation for more advanced algebra.", "---", "Keywords:\n(\sqrt{(b-3)^2} = 5), solve absolute value equations, solve (\sqrt{x^2} = a), absolute value definition, algebra practice, step-by-step equation solving, real-world math applications", "---", "Whether you're a student preparing for exams or someone brushing up on algebra basics, mastering this concept ensures confidence when encountering similar equations. Always remember: the square root of a square equals the absolute value—this rule simplifies solving equations involving square roots of squared expressions."]
