\( 12^2 + b^2 = 20^2 \) → \( 144 + b^2 = 400 \) → \( b^2 = 256 \) → \( b = 16 \)

["Solving the Equation ( 12^2 + b^2 = 20^2 ): A Step-by-Step Guide", "Mathematics often involves solving equations to uncover hidden values — whether in algebra, geometry, or real-world applications. One classic example is the equation:", "[\n12^2 + b^2 = 20^2\n]", "This equation presents a Pythagorean twist, where (12) and (b) form two legs of a right triangle, and (20) is the hypotenuse. Let’s break down how to solve it step by step, demonstrating both algebraic techniques and practical understanding.", "---", "### Step 1: Recognize the Pythagorean Relationship", "At first glance, (12^2 + b^2 = 20^2) resembles the Pythagorean theorem:", "[\na^2 + b^2 = c^2\n]", "Here, (c = 20), one leg (= 12), and the unknown leg (b) is what we’re solving for. The equation simplifies neatly:", "[\n144 + b^2 = 400\n]", "---", "### Step 2: Isolate the Variable Term", "To find (b^2), subtract (144) from both sides:", "[\nb^2 = 400 - 144 = 256\n]", "This subtraction yields:", "[\nb^2 = 256\n]", "---", "### Step 3: Solve for (b) by Taking the Square Root", "Now, take the square root of both sides to isolate (b):", "[\nb = \sqrt{256} = 16\n]", "Note: Since (b) represents a length (as in geometric problems), we consider only the positive root.", "---", "### Why This Matters: Applications of Pythagorean Triples", "The values (12), (16), and (20) form a well-known Pythagorean triple. Pythagorean triples — sets of integers satisfying (a^2 + b^2 = c^2) — are essential in geometry, architecture, and physics for understanding right triangles and distances.", "---", "### Conclusion", "Solving the equation (12^2 + b^2 = 20^2) leads logically to:", "[\nb = 16\n]", "Understanding this process enhances problem-solving skills in algebra and reinforces the elegance of fundamental mathematical principles. Whether you’re working on homework, preparing for exams, or exploring STEM fields, mastering equations like this builds a solid foundation for more advanced topics.", "---", "Keywords: (12^2 + b^2 = 20^2), solve for (b), Pythagorean theorem, algebra steps, (b = 16), solving quadratic equations, geometry algebra, math problem-solving.", "---", "By breaking down each step clearly, you gain confidence in manipulating algebraic expressions and recognizing geometric insights in equations. Keep practicing — every equation is a puzzle waiting to be solved!"]