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

["Solving the Equation: \frac{2a^2 + 2b^2}{a^2 - b^2} = 3 – A Step-by-Step Guide", "Understanding algebraic equations is fundamental in mathematics, and equations like \frac{2a^2 + 2b^2}{a^2 - b^2} = 3 appear frequently in algebra, calculus, and applied sciences. In this article, we’ll break down how to solve and interpret this equation, explore its algebraic meaning, and guide you through finding real values for (a) and (b).", "---", "### What Is the Equation?", "We start with:", "[\n\frac{2a^2 + 2b^2}{a^2 - b^2} = 3\n]", "This equation equates a rational expression—two quadratic expressions in the numerator and denominator—to the constant 3. Our goal is to solve for (a) and (b), under the condition that the denominator is non-zero.", "---", "### Step 1: Simplify the Expression", "Factor out constants in the numerator:", "[\n\frac{2(a^2 + b^2)}{a^2 - b^2} = 3\n]", "Notice that (a^2 - b^2) is a difference of squares, so rewrite the denominator:", "[\na^2 - b^2 = (a - b)(a + b)\n]", "Thus, the equation becomes:", "[\n\frac{2(a^2 + b^2)}{(a - b)(a + b)} = 3\n]", "---", "### Step 2: Eliminate the Denominator", "Multiply both sides by (a^2 - b^2) (assuming (a^2 <br/>\ne b^2) to keep the expression defined):", "[\n2(a^2 + b^2) = 3(a^2 - b^2)\n]", "Now expand both sides:", "[\n2a^2 + 2b^2 = 3a^2 - 3b^2\n]", "Bring all terms to one side to collect like terms:", "[\n2a^2 + 2b^2 - 3a^2 + 3b^2 = 0\n]", "Simplify:", "[\n(-a^2 + 5b^2) = 0\n]", "So,", "[\na^2 = 5b^2\n]", "---", "### Step 3: Solve for (a) in Terms of (b)", "Take square roots on both sides:", "[\na = \pm\sqrt{5}b\n]", "Important Note: Since (a^2 - b^2 <br/>\ne 0), we require (a^2 <br/>\ne b^2), which holds true because (\sqrt{5}^2 = 5 <br/>\ne 1 = b^2), assuming (b <br/>\ne 0). If (b = 0), then (a = 0), but this makes both numerator and denominator zero—undesirable as the expression becomes indeterminate. So, (b <br/>\ne 0) is required.", "---", "### Practical Interpretation and Solutions", "The solution to the equation is:", "[\na = \sqrt{5},b \quad \ ext{or} \quad a = -\sqrt{5},b, \quad \ ext{with } b <br/>\ne 0\n]", "This represents two families of real solutions parameterized by (b), avoiding the undefined point where (a^2 = b^2).", "For example, if (b = 1), then (a = \sqrt{5}) or (a = -\sqrt{5}). Plugging back confirms these values satisfy the original equation.", "---", "### Applications and Why It Matters", "Equations of this form appear in fields such as:", "- Physics: Analyzing motion or forces involving quadratic terms.\n- Engineering: Modeling stress-strain relationships or circuit analysis.\n- Economics: Deriving ratios in optimization problems.\n- Graphing: Understanding asymptotes and domain restrictions.", "Knowing how to isolate variables and reduce expressions is vital for solving more complex models.", "---", "### Final Tips", "- Always check that your solution doesn’t make the denominator zero.\n- Simplify using factoring and properties of exponents.\n- Express solutions in parametric form for broader applicability.\n- Test values if unsure about the domain.", "---", "Conclusion\nThe equation \frac{2a^2 + 2b^2}{a^2 - b^2} = 3 leads cleanly to the relationship (a^2 = 5b^2), with solutions (a = \pm\sqrt{5},b), under the condition (b <br/>\ne 0). Mastering this type of algebraic manipulation strengthens your problem-solving toolkit across mathematics and science.", "---", "Related Searches:", "- Solve (\frac{2a^2 + 2b^2}{a^2 - b^2} = k)\n- How to solve rational equations\n- Algebraic simplification techniques\n- Difference of squares applications\n- Solving quadratic ratios", "---", "Keywords: (\frac{2a^2 + 2b^2}{a^2 - b^2} = 3), algebraic equation, solve for (a) and (b), rational expression, difference of squares, mathematical solution guide"]