2r - (3 - \sqrt{5})r + 5 - \sqrt{5} = 0
["Title: Solve the Equation 2r − (3 − √5)r + 5 − √5 = 0: Step-by-Step Exact Solution", "---", "Introduction\nSolving quadratic equations can be challenging, especially when they involve irrational numbers like √5. In this article, we will carefully solve the equation:\n2r − (3 − √5)r + 5 − √5 = 0, using mathematical steps to find precise, exact solutions. Whether you're a student, teacher, or self-learner, this guide will help you understand how to simplify and solve quadratic expressions with √5.", "---", "### Step 1: Understand the Equation Structure", "The given equation is:\n2r − (3 − √5)r + 5 − √5 = 0", "We notice two main terms involving r and a constant term:\n- Coefficient of r: (2 − (3 − √5))\n- Constant term: (5 − √5)", "We rewrite it in standard quadratic form:\nar² + br + c = 0, where\n- ( a = 2 - (3 - \sqrt{5}) )\n- ( b = 0 ) (no r² term)\n- ( c = 5 - \sqrt{5} )", "Since the coefficient of r² is zero but there is a linear term and constant, this simplifies to a linear equation in disguise. However, let’s verify and solve carefully.", "---", "### Step 2: Simplify the Coefficient of r", "Simplify the expression for the coefficient of r:\n[\na = 2 - (3 - \sqrt{5}) = 2 - 3 + \sqrt{5} = -1 + \sqrt{5}\n]", "The equation becomes:\n[\n(-1 + \sqrt{5})r + (5 - \sqrt{5}) = 0\n]", "---", "### Step 3: Solve for r", "Isolate r by moving the constant to the other side:\n[\n(-1 + \sqrt{5})r = - (5 - \sqrt{5})\n]\n[\n(-1 + \sqrt{5})r = -5 + \sqrt{5}\n]", "Now divide both sides by ( -1 + \sqrt{5} ):\n[\nr = \frac{-5 + \sqrt{5}}{-1 + \sqrt{5}}\n]", "---", "### Step 4: Rationalize the Denominator", "We rationalize the denominator ( -1 + \sqrt{5} ) by multiplying numerator and denominator by its conjugate ( -1 - \sqrt{5} ):\n[\nr = \frac{(-5 + \sqrt{5})(-1 - \sqrt{5})}{(-1 + \sqrt{5})(-1 - \sqrt{5})}\n]", "Denominator:\n[\n(-1)^2 - (\sqrt{5})^2 = 1 - 5 = -4\n]", "Numerator:\nUse distributive property (FOIL):\n[\n(-5)(-1) + (-5)(-\sqrt{5}) + (\sqrt{5})(-1) + (\sqrt{5})(-\sqrt{5}) = 5 + 5\sqrt{5} - \sqrt{5} - 5\n]\nSimplify:\n[\n(5 - 5) + (5\sqrt{5} - \sqrt{5}) = 4\sqrt{5}\n]", "Now combine:\n[\nr = \frac{4\sqrt{5}}{-4} = -\sqrt{5}\n]", "---", "### Step 5: Final Answer", "The exact solution to the equation is:\nr = –√5", "---", "### Bonus: Verify the Solution", "Plug ( r = -\sqrt{5} ) back into the original equation:\nLeft-hand side:\n[\n2(-\sqrt{5}) - (3 - \sqrt{5})(-\sqrt{5}) + 5 - \sqrt{5}\n]\nCalculate each term:\n- ( 2(-\sqrt{5}) = -2\sqrt{5} )\n- ( (3 - \sqrt{5})(-\sqrt{5}) = -3\sqrt{5} + 5 ), so minus that becomes ( +3\sqrt{5} - 5 )\n- Constant: ( +5 - \sqrt{5} )", "Add all together:\n[\n-2\sqrt{5} + 3\sqrt{5} - 5 + 5 - \sqrt{5} = (-2 + 3 - 1)\sqrt{5} + (-5 + 5) = 0 + 0 = 0\n]", "✅ Verified!", "---", "### Why This Matters", "Equations like ( 2r - (3 - \sqrt{5})r + 5 - \sqrt{5} = 0 ) model real-world scenarios involving irrational parameters—common in physics, engineering, and economics. Understanding how to solve such equations builds a strong foundation in algebra with irrational numbers.", "---", "Keywords:\nsolve 2r − (3 − √5)r + 5 − √5 = 0, quadratic equation with √5, algebra solution, rationalizing denominator, exact root, irrational number equation, step-by-step math tutorial, solve linear with radicals", "---", "About the Author:\nMathWise Education specializes in clear, accurate explanations of algebraic concepts. Understanding equations with radicals starts here—master the basics to tackle advanced math with confidence.", "---", "Learn more about solving quadratic expressions, irrational numbers, and algebraic manipulation at [yourwebsite.com]"]
