x^2 + \frac{9}{16}x^2 = 25

["# Solving the Equation: x² + (\frac{9}{16})x² = 25", "If you're tackling the algebraic equation x² + (\frac{9}{16})x² = 25, you're stepping into a classic quadratic-style simplification problem built on combining like terms. This seemingly straightforward equation actually hides a powerful method for solving linearCombined quadratic expressions — a skill widely used in algebra, physics, and engineering. In this article, we’ll walk through solving this equation step-by-step, explore its structure, and explain how it connects to real-world applications.", "---", "## Step-by-Step Solution of the Equation", "### Step 1: Combine Like Terms\nThe left-hand side of the equation contains two terms with x²:\n[ x^2 + \frac{9}{16}x^2 ]", "To simplify, factor out x²:\n[ x^2 \left(1 + \frac{9}{16}\right) ]", "Now calculate the sum inside the parentheses:\n[ 1 + \frac{9}{16} = \frac{16}{16} + \frac{9}{16} = \frac{25}{16} ]", "So the equation becomes:\n[ \frac{25}{16}x^2 = 25 ]", "---", "### Step 2: Isolate the Variable", "To solve for x², divide both sides by (\frac{25}{16}):\n[ x^2 = 25 \div \frac{25}{16} ]", "Recall that dividing by a fraction is the same as multiplying by its reciprocal:\n[ x^2 = 25 \ imes \frac{16}{25} = 16 ]", "---", "### Step 3: Solve for x", "Now that you have:\n[ x^2 = 16 ]", "Take the square root of both sides:\n[ x = \pm \sqrt{16} ]\n[ x = \pm 4 ]", "---", "## Final Answer", "The solutions to the equation x² + (\frac{9}{16})x² = 25 are:\n[ \boxed{x = 4} \quad \ ext{and} \quad \boxed{x = -4} ]", "---", "## Why This Equation Matters: Combining Like Terms", "Combining like terms is essential in simplifying algebraic expressions, especially in complex equations. When multiple terms involve the same variable raised to the same power, combining them reduces complexity and reveals underlying relationships — crucial for solving nonlinear equations efficiently.", "This type of simplification often appears in:\n- Physics, when balancing energy equations with fractional coefficients\n- Engineering, when modeling systems with combined response ratios\n- Economics, when aggregating proportional changes", "---", "## How to Solve It: Quick Record", "| Step | Operation | Result |\n|-------------------|-------------------------------|-------------------|\n| 1 | Combine x² terms | (\frac{25}{16}x^2) |\n| 2 | Divide both sides by (\frac{25}{16}) | (x^2 = 16) |\n| 3 | Take square root | (x = \pm 4) |", "---", "## Summary", "Solving x² + (\frac{9}{16})x² = 25 hinges on recognizing and combining like terms, simplifying the equation into a solvable quadratic form:\n[ \frac{25}{16}x^2 = 25 \Rightarrow x^2 = 16 \Rightarrow x = \pm 4 ]", "Understanding this process strengthens your algebraic foundation and prepares you for more advanced mathematical challenges across science and technology fields.", "---", "## FAQ: Common Questions About This Equation", "Q: Why do we combine like terms first?\nA: Simplifying expressions reduces errors and makes solving equations faster and clearer.", "Q: Does this equation represent a real-world scenario?\nA: Yes! It models situations involving proportional scales, such as dilutions in chemistry or scaled proportional growth.", "Q: What if I get a fraction under the root? How do I simplify?\nA: Use the property ( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} ), or multiply numerator and denominator by √b to rationalize.", "---", "Explore similar problems and deepen your algebra skills by mastering how to combine terms, simplify quadratic expressions, and solve for variables efficiently!"]