Solving for \( b^2 \) gives \( b^2 = 144 \), so \( b = 12 \).

["# How Solving for ( b^2 = 144 ) Gives ( b = 12 ): Explaining the Step-by-Step Path to the Solution", "Solving quadratic equations is a fundamental skill in mathematics, and understanding how values like ( b^2 = 144 ) lead to definitive answers—such as ( b = 12 )—is crucial for students and learners alike. In this article, we’ll break down the process behind the equation ( b^2 = 144 ) and why its solution resolves cleanly to ( b = 12 ).", "## Understanding the Equation: ( b^2 = 144 )", "At its core, ( b^2 = 144 ) means "b squared equals 144." This equation states that ( b ) multiplied by itself equals 144. To find the value(s) of ( b ), we need to determine which number—when squared—equals 144.", "### Taking the Square Root of Both Sides", "To solve for ( b ), we apply the square root function to both sides of the equation:\n[\n\sqrt{b^2} = \sqrt{144}\n]", "The left-hand side simplifies as ( \sqrt{b^2} = |b| ), meaning the absolute value of ( b ). This is important because squaring a number makes the result positive, so both positive and negative values of ( b ) satisfy the original equation. Therefore:\n[\n|b| = 12\n]", "### Why Absolute Value Matters", "Since squaring both positive and negative values yields the same result, both ( b = 12 ) and ( b = -12 ) satisfy ( b^2 = 144 ). Thus, the complete solution set is:\n[\nb = \pm12\n]", "However, if the problem context specifies that ( b ) is positive—such as in geometric or practical applications—we often report only the positive root. So, in many scenarios, the answer given is ( b = 12 ).", "## The Clear Solution: ( b = 12 )", "Because ( 12^2 = 144 ) (since ( 12 \ imes 12 = 144 )), it follows that:\n[\nb^2 = 144 \implies b = 12\n]", "This straightforward result reveals that 12 is the primary positive solution to the equation. While ( b = -12 ) is mathematically valid, such details are typically emphasized only when context permits.", "## Step-by-Step Summary", "Here’s a quick recap of the logical pathway:\n1. Start with ( b^2 = 144 )\n2. Apply square root: ( \sqrt{b^2} = \sqrt{144} ) → ( |b| = 12 )\n3. Solve for ( b ): ( b = \pm12 )\n4. Select ( b = 12 ) if context assumes positivity", "## Real-World Implications", "The equation ( b^2 = 144 ) appears across physics, engineering, and finance—especially in formulas involving distances, velocities, or area calculations. Recognizing when ( b = 12 ) (or both ( \pm12 )) allows accurate problem-solving and insight into mathematical relationships that define real-world phenomena.", "## Final Thoughts", "Solving ( b^2 = 144 ) demonstrates how algebraic manipulation leads directly to clear solutions. Understanding that both positive and negative roots exist, yet selecting the positive value in context simplifies communication and interpretation. By mastering this process, learners build a solid foundation for tackling more complex equations and applications.", "Key takeaway:\nGiven ( b^2 = 144 ), solving step by step leads us to ( b = 12 )—a fundamental result that highlights the power of balance and symmetry in mathematics.", "---\nThis structured approach not only solves the equation but also clarifies why ( b = 12 ) is the valid solution in most real-world settings. Whether studying algebra, preparing for exams, or solving applied problems, understanding this process strengthens mathematical reasoning and problem-solving skills."]