\frac{a^2 + b^2}{a^2 - b^2} = \frac{3}{2}

["Solving the Equation: \frac{a² + b²}{a² - b²} = \frac{3}{2} – A Step-by-Step Guide", "Mathematics often presents challenging equations that require a logical approach to solve. One such equation is:", "[\n\frac{a^2 + b^2}{a^2 - b^2} = \frac{3}{2}\n]", "This equation involves quadratic terms in (a^2) and (b^2) and can be solved algebraically to find the relationship between (a) and (b). In this article, we’ll break down the solution step by step and explore its significance in algebra, geometry, and real-world applications.", "---", "### Step 1: Eliminate the denominator", "To eliminate the fractions, multiply both sides of the equation by (a^2 - b^2) (assuming (a^2 <br/>\ne b^2)):", "[\na^2 + b^2 = \frac{3}{2}(a^2 - b^2)\n]", "---", "### Step 2: Expand the right-hand side", "Distribute (\frac{3}{2}):", "[\na^2 + b^2 = \frac{3}{2}a^2 - \frac{3}{2}b^2\n]", "---", "### Step 3: Collect like terms", "Bring all terms involving (a^2) and (b^2) to one side. Subtract (a^2) and add (\frac{3}{2}b^2) to both sides:", "[\na^2 - \frac{3}{2}a^2 + b^2 + \frac{3}{2}b^2 = 0\n]", "Simplify the coefficients:", "[\n-\frac{1}{2}a^2 + \frac{5}{2}b^2 = 0\n]", "---", "### Step 4: Multiply both sides by 2 to eliminate fractions", "[\n- a^2 + 5b^2 = 0\n]", "---", "### Step 5: Rearrange the equation", "[\na^2 = 5b^2\n]", "---", "### Step 6: Express the ratio or linear relationship", "Take square roots (noting (a, b) are real, so consider positive roots for clarity):", "[\na = \sqrt{5},b\n\quad \ ext{or} \quad\na = -\sqrt{5},b\n]", "This shows that (a) and (b) are in a fixed ratio proportional to (\sqrt{5}), valid when (a^2 <br/>\ne b^2), as the original denominator (a^2 - b^2) must not be zero.", "---", "### Key Insights and Applications", "- Rational Expressions: This problem illustrates solving rational equations involving quadratic expressions—common in algebra exams and standardized tests.", "- Geometric Interpretation: In coordinate geometry, ratios like (\frac{a^2 + b^2}{a^2 - b^2} = \frac{3}{2}) can appear when analyzing distances, slopes, or similar triangles.", "- Real World Context: Such equations model phenomena in physics or engineering where squared quantities relate to energy, power ratios, or signal processing.", "---", "### Summary", "The equation (\frac{a^2 + b^2}{a^2 - b^2} = \frac{3}{2}) simplifies neatly to (a^2 = 5b^2), revealing a proportional relationship between (a) and (b). This algebraic solution strengthens problem-solving skills applicable across mathematical disciplines.", "---", "Need more algebra help? Whether solving equations or exploring deeper concepts, understanding rational expressions unlocks powerful tools in mathematics and science.", "---", "Keywords: (\frac{a^2 + b^2}{a^2 - b^2} = \frac{3}{2}), solving rational equations, algebra, quadratic expressions, mathematica tutorial, equation solving, math tips."]