Case 2:** \( rac{r+1}{r-1} = rac{3 - \sqrt{5}}{2} \)

["# Case 2: Solving the Equation ( \dfrac{r+1}{r-1} = \dfrac{3 - \sqrt{5}}{2} )", "When tackling algebraic equations in exams or self-study, equations like\nCase 2: ( \dfrac{r+1}{r-1} = \dfrac{3 - \sqrt{5}}{2} )\ncan seem challenging at first, but with a systematic approach, they become manageable. This article walks you through solving this rational equation step-by-step, explains the reasoning behind each step, and offers practical applications to strengthen your algebra skills.", "---", "## Problem Overview", "We are given the equation:\n[\n\frac{r + 1}{r - 1} = \frac{3 - \sqrt{5}}{2}\n]\nGoal: Solve for ( r ) in terms of radicals.", "This equation is of Case 2 in standard comparison problems, where one rational expression is set equal to an irrational constant. Solving such equations requires cross-multiplication and algebraic manipulation to isolate the unknown variable.", "---", "## Step-by-Step Solution", "### Step 1: Cross-multiply and simplify\nStart by multiplying both sides of the equation by ( r - 1 ) to eliminate the denominator on the left:\n[\nr + 1 = \left( \dfrac{3 - \sqrt{5}}{2} \right)(r - 1)\n]", "### Step 2: Expand the right-hand side\nDistribute the constant on the right-hand side:\n[\nr + 1 = \left( \dfrac{3 - \sqrt{5}}{2} \right)r - \left( \dfrac{3 - \sqrt{5}}{2} \right)\n]", "### Step 3: Move all terms involving ( r ) to one side\nSubtract ( \left( \dfrac{3 - \sqrt{5}}{2} \right)r ) from both sides:\n[\nr - \left( \dfrac{3 - \sqrt{5}}{2} \right)r + 1 = - \left( \dfrac{3 - \sqrt{5}}{2} \right)\n]", "Factor ( r ):\n[\nr\left( 1 - \dfrac{3 - \sqrt{5}}{2} \right) + 1 = - \dfrac{3 - \sqrt{5}}{2}\n]", "Simplify the coefficient of ( r ):\n[\n1 = \dfrac{2}{2}, \quad \ ext{so} \quad \dfrac{2 - (3 - \sqrt{5})}{2} = \dfrac{2 - 3 + \sqrt{5}}{2} = \dfrac{-1 + \sqrt{5}}{2}\n]", "Thus:\n[\nr \cdot \dfrac{-1 + \sqrt{5}}{2} + 1 = - \dfrac{3 - \sqrt{5}}{2}\n]", "### Step 4: Isolate ( r )\nSubtract 1 from both sides:\n[\nr \cdot \dfrac{-1 + \sqrt{5}}{2} = - \dfrac{3 - \sqrt{5}}{2} - 1\n]", "Convert 1 to half:\n[\n-1 = - \dfrac{2}{2}, \quad \ ext{so} \quad - \dfrac{3 - \sqrt{5}}{2} - \dfrac{2}{2} = \dfrac{-3 + \sqrt{5} - 2}{2} = \dfrac{-5 + \sqrt{5}}{2}\n]", "Now:\n[\nr \cdot \dfrac{-1 + \sqrt{5}}{2} = \dfrac{-5 + \sqrt{5}}{2}\n]", "Multiply both sides by 2:\n[\nr(-1 + \sqrt{5}) = -5 + \sqrt{5}\n]", "### Step 5: Solve for ( r )\nDivide both sides by ( \sqrt{5} - 1 ):\n[\nr = \dfrac{-5 + \sqrt{5}}{\sqrt{5} - 1}\n]", "Multiply numerator and denominator by the conjugate of the denominator, ( \sqrt{5} + 1 ), to rationalize:\n[\nr = \dfrac{(-5 + \sqrt{5})(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)}\n]", "Compute denominator:\n[\n(\sqrt{5})^2 - (1)^2 = 5 - 1 = 4\n]", "Compute numerator:\n[\n(-5)(\sqrt{5}) + (-5)(1) + (\sqrt{5})(\sqrt{5}) + (\sqrt{5})(1) = -5\sqrt{5} - 5 + 5 + \sqrt{5} = (-5\sqrt{5} + \sqrt{5}) + (-5 + 5) = -4\sqrt{5}\n]", "So:\n[\nr = \dfrac{-4\sqrt{5}}{4} = -\sqrt{5}\n]", "---", "## Verification", "Plug ( r = -\sqrt{5} ) back into the original equation:\n[\n\frac{-\sqrt{5} + 1}{-\sqrt{5} - 1} = \frac{1 - \sqrt{5}}{-(\sqrt{5} + 1)} = \frac{-( \sqrt{5} - 1 )}{-(\sqrt{5} + 1)} = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\n]", "Now rationalize the right-hand side:\n[\n\frac{\sqrt{5} - 1}{\sqrt{5} + 1} \cdot \frac{\sqrt{5} - 1}{\sqrt{5} - 1} = \frac{(\sqrt{5} - 1)^2}{5 - 1} = \frac{5 - 2\sqrt{5} + 1}{4} = \frac{6 - 2\sqrt{5}}{4} = \frac{3 - \sqrt{5}}{2}\n]", "Which matches the right-hand side. Verification successful!", "---", "## Real-World and Mathematical Significance", "Equations like this appear in physics (e.g., modeling resonance in oscillatory systems), finance (e.g., growth models with nonlinear terms), and geometry (e.g., solving proportions involving irrational lengths). Solving such equations builds foundational skills for:", "- Rational function manipulation\n- Pattern recognition in algebraic forms\n- Applying rationalization and conjugate techniques\n- Checking solutions for consistency", "---", "## Alternative Perspectives", "The right-hand side ( \dfrac{3 - \sqrt{5}}{2} ) arises naturally in so-called golden mean-related expressions, often seen in recursive sequences and quadratic surds. Understanding how to solve rational equations involving radicals equips you to tackle deeper problems in number theory, continued fractions, and dynamical systems.", "---", "## Conclusion", "Solving Case 2 equations like\n[\n\dfrac{r+1}{r-1} = \dfrac{3 - \sqrt{5}}{2}\n]\ndemands careful cross-multiplication, coefficient management, and strategic rationalization. Mastering these steps enhances algebraic fluency and prepares you to handle complex, real-world problems where exact solutions are critical.", "💡 Tip: Always verify your solution by substitution — especially when radicals and fractions are involved!", "---", "Keywords for SEO: Case 2 algebra, solving rational equations, algebraic solutions with radicals, solving (r+1)/(r−1) = (3−√5)/2, step-by-step radical equation, irrational equation solving, algebra tutorial case 2."]