\(25 + h^2 = 169\) → \(h^2 = 144\) → \(h = 12\) feet.

["# Solving the Equation (25 + h^2 = 169): A Step-by-Step Guide", "Mathematics is a language of problem-solving, and equations like (25 + h^2 = 169) offer a clear pathway to finding unknown variables. Whether you're a student learning algebra or someone brushing up on fundamental math skills, understanding how to solve such equations is essential. In this article, we’ll walk through solving (25 + h^2 = 169) step by step, arriving at the solution (h = 12) feet. This foundational skill applies not only in academic settings but also in real-world applications like physics, engineering, and design.", "## Understanding the Original Equation", "The equation (25 + h^2 = 169) is a simple quadratic expression. It states that when you add 25 to the square of (h), the total equals 169. The goal is to isolate (h) and determine its value.", "## Step 1: Subtract 25 from Both Sides", "To begin isolating (h^2), subtract 25 from both sides:", "[\n25 + h^2 - 25 = 169 - 25\n]", "Simplifying both sides gives:\n[\nh^2 = 144\n]", "This step removes the constant, leaving the square of (h) on the left.", "## Step 2: Take the Square Root of Both Sides", "Now that (h^2 = 144), we take the square root of both sides to solve for (h). Remember:\n[\n\sqrt{h^2} = |h|\n]\nso the equation technically gives two possible solutions:", "[\nh = \pm\sqrt{144} \Rightarrow h = \pm12\n]", "However, in many practical contexts—especially when (h) represents a physical dimension like height or distance—only the positive value is meaningful. Since (h) is stated to be 12 feet, we accept:", "[\nh = 12 \ ext{ feet}\n]", "## Why the Absolute Value Matters", "When solving equations involving squares, remember that both positive and negative roots are valid mathematically. But context often determines which solution fits real-world use. Here, because (h) represents height (a non-negative quantity), we conclude (h = 12) feet without ambiguity.", "## Real-World Applications of This Equation", "Equations of the form (h^2 + C = D) appear frequently in physics and engineering. For example:", "- Calculating distance from displacement in kinematics\n- Determining electrical resistance in parallel circuits\n- Computing energy in mechanical systems", "Understanding how to isolate variables and interpret solutions empowers you to apply algebra confidently across disciplines.", "## Summary", "Solving (25 + h^2 = 169) involves:\n1. Subtracting 25 to isolate (h^2), resulting in (h^2 = 144).\n2. Taking the square root to find (h = \pm12).\n3. Selecting (h = 12) feet based on practical context.", "Mastering such basics strengthens your mathematical foundation and enhances problem-solving skills in diverse fields.", "---", "### Key Takeaways", "- Always isolate the variable term first.\n- Remember both positive and negative square roots algebraically.\n- Context guides which solution to accept in real applications.\n- Practice layering steps methodically for accuracy.", "Whether you’re solving for (h) in a textbook or applying algebra in a lab or design project, the principles remain the same: clarity, precision, and informed judgment define successful math problem-solving.", "---", "Want to master algebra? Explore more about solving quadratic equations, order of operations, and real-world math applications today!"]