Simplifying gives \( 25 + b^2 = 169 \).

["Simplifying the Equation: How to Solve (25 + b^2 = 169)", "Solving quadratic equations is a fundamental skill in algebra, and simplifying equations step-by-step makes them much easier to manage. One common problem students encounter is simplifying and solving the equation:", "[\n25 + b^2 = 169\n]", "In this article, we’ll walk through how to simplify this equation, solve for ( b ), and understand the key steps involved.", "---", "### Step 1: Isolate the ( b^2 ) Term", "Start by eliminating the constant term on the left side. Subtract 25 from both sides of the equation:", "[\n25 + b^2 - 25 = 169 - 25\n]", "Simplifying both sides gives:", "[\nb^2 = 144\n]", "---", "### Step 2: Solve for ( b ) by Taking the Square Root", "To find ( b ), take the square root of both sides:", "[\nb = \pm\sqrt{144}\n]", "Since ( \sqrt{144} = 12 ), the solution becomes:", "[\nb = \pm12\n]", "This means there are two solutions: ( b = 12 ) and ( b = -12 ).", "---", "### Why This Simplification Works", "The key to simplifying ( 25 + b^2 = 169 ) lies in recognizing that the equation follows standard quadratic form when rearranged properly. Subtracting 25 clears the constant term, leaving just ( b^2 ), which allows direct square root elimination.", "---", "### Bonus: Check Your Solutions", "Always verify by substituting back into the original equation:", "- For ( b = 12 ):\n ( 25 + (12)^2 = 25 + 144 = 169 ) ✅\n- For ( b = -12 ):\n ( 25 + (-12)^2 = 25 + 144 = 169 ) ✅", "Both values satisfy the equation.", "---", "### Summary", "Simplifying ( 25 + b^2 = 169 ) involves isolating ( b^2 ) and solving using square roots, resulting in two real solutions. By carefully subtracting constants and applying square root properties, even square root equations become straightforward. This method is a clear example of how basic algebraic manipulation leads to straightforward solutions.", "---", "Key Takeaways:", "- Subtract constants to simplify the equation before solving for ( b^2 ).\n- Take square roots carefully—remember solutions are ( \pm \sqrt{} ).\n- Always verify solutions in the original equation.", "Mastering these steps not only helps solve this specific problem but builds a solid foundation for tackling more complex quadratic equations.", "---", "If you found this guide helpful, explore more algebra basics and quadratic equation techniques to boost your math confidence!"]