That simplifies to \( 81 + b^2 = 225 \). Subtracting 81 from both sides, \( b^2 = 144 \).

["# Simplifying the Equation: Solving ( 81 + b^2 = 225 )", "Mathematics often involves taking complex-looking expressions and simplifying them into clear, solvable forms. One classic example is the equation:", "[\n81 + b^2 = 225\n]", "This equation simplified step-by-step helps students and learners understand the fundamentals of algebra — particularly the process of isolating the variable term. Let’s break it down.", "---", "### Step 1: Subtract 81 from Both Sides", "The first essential algebraic step is to isolate ( b^2 ). To do this, subtract 81 from both sides of the equation to eliminate the constant term on the left:", "[\n81 + b^2 - 81 = 225 - 81\n]", "Simplifying both sides gives:", "[\nb^2 = 144\n]", "This step is fundamental because it removes all other terms from one side, leaving a single expression containing the variable.", "---", "### Step 2: Solve for ( b ) by Taking the Square Root", "Now that we have ( b^2 = 144 ), we solve for ( b ) by applying the principle that taking the square root of both sides gives both positive and negative solutions:", "[\nb = \pm \sqrt{144}\n]", "Since the square root of 144 is 12, we find:", "[\nb = \pm 12\n]", "This means the solutions are:", "[\nb = 12 \quad \ ext{or} \quad b = -12\n]", "---", "### Why This Equation Matters", "This type of quadratic equation — a simple form of ( x^2 = c ) — is foundational in algebra. It teaches learners how to simplify, isolate variables, and recognize perfect squares. It also serves as stepping stone to more complex equations involving cubes, polynomials, and real-world modeling.", "Understanding how to simplify equations by subtracting constants and solving for variables is not just about finding answers — it’s about mastering logical thinking and step-by-step problem-solving.", "---", "### Summary", "- Start with: ( 81 + b^2 = 225 )\n- Subtract 81: ( b^2 = 144 )\n- Solve: ( b = \pm 12 )", "Whether you’re a student learning algebra or someone brushing up on fundamentals, simplifying equations step-by-step empowers clearer thinking and stronger math skills. Keep practicing — each equation brings you one step closer to mastery.", "---", "### Additional Tips", "- Always balance equations by performing the same operation on both sides.\n- Recognizing perfect squares simplifies solutions dramatically.\n- Don’t forget to consider both positive and negative roots.\n- Use tools like squares charts or square root properties to verify solutions.", "---", "If you found this explanation helpful, share it with fellow learners — mastering algebra starts with understanding simples like ( b^2 = 144 ). Keep simplifying!"]