2r + 2 = (3 + \sqrt{5})r - 3 - \sqrt{5}

["# Solving the Equation 2r + 2 = (3 + √5)r − 3 − √5: A Step-by-Step Guide", "In algebra, solving equations accurately is essential for understanding mathematical relationships. One intriguing equation often studied comes in the form:", "$$\n2r + 2 = (3 + \sqrt{5})r - 3 - \sqrt{5}\n$$", "This article breaks down how to solve this equation step-by-step, uncovering the value of ( r ) and highlighting key algebraic techniques. Whether you're a student, educator, or math enthusiast, mastering this problem strengthens your problem-solving skills.", "---", "## Step 1: Understand the Equation Structure", "We begin with the equation:", "$$\n2r + 2 = (3 + \sqrt{5})r - 3 - \sqrt{5}\n$$", "This is a linear equation involving a radical (( \sqrt{5} )), meaning the variable ( r ) appears both linearly and multiplied by ( \sqrt{5} ). The presence of irrational numbers suggests the solution may involve ( \sqrt{5} ), but careful algebraic manipulation eliminates these terms.", "---", "## Step 2: Rearranging Terms to Isolate ( r )", "First, bring all terms involving ( r ) to one side and constant terms to the other. Subtract ( 2r ) from both sides and add ( 3 + \sqrt{5} ) to both sides:", "$$\n2r - (3 + \sqrt{5})r = -3 - \sqrt{5} - 2\n$$", "Simplify the right-hand side:", "$$\n(2 - (3 + \sqrt{5}))r = -5 - \sqrt{5}\n$$", "Compute the coefficient of ( r ):", "$$\n(2 - 3 - \sqrt{5})r = -1 - \sqrt{5}\n$$", "So:", "$$\n(-1 - \sqrt{5})r = -5 - \sqrt{5}\n$$", "---", "## Step 3: Solving for ( r )", "Now divide both sides by ( -1 - \sqrt{5} ):", "$$\nr = \frac{-5 - \sqrt{5}}{-1 - \sqrt{5}}\n$$", "To simplify, multiply numerator and denominator by the conjugate of the denominator, ( -1 + \sqrt{5} ), to eliminate the radical in the denominator:", "$$\nr = \frac{(-5 - \sqrt{5})(-1 + \sqrt{5})}{(-1 - \sqrt{5})(-1 + \sqrt{5})}\n$$", "---", "## Step 4: Simplify the Numerator and Denominator", "Denominator:\nThis is a difference of squares:", "$$\n(-1)^2 - (\sqrt{5})^2 = 1 - 5 = -4\n$$", "Numerator:\nUse distributive property:", "$$\n(-5 - \sqrt{5})(-1 + \sqrt{5}) = (-5)(-1) + (-5)(\sqrt{5}) + (-\sqrt{5})(-1) + (-\sqrt{5})(\sqrt{5})\n$$", "$$\n= 5 - 5\sqrt{5} + \sqrt{5} - 5 = (5 - 5) + (-5\sqrt{5} + \sqrt{5}) = -4\sqrt{5}\n$$", "Thus, numerator simplifies to ( -4\sqrt{5} ).", "---", "## Step 5: Final Division", "$$\nr = \frac{-4\sqrt{5}}{-4} = \sqrt{5}\n$$", "---", "## Conclusion", "The solution to the equation\n$$\n2r + 2 = (3 + \sqrt{5})r - 3 - \sqrt{5}\n$$\nis\n$$\n\boxed{r = \sqrt{5}}\n$$", "This problem illustrates how rationalizing the denominator and careful manipulation of radicals enable simplification of equations involving irrational coefficients. Knowing how to solve such equations is valuable not only in academic settings but also in applied fields like engineering and physics, where irrational constants naturally arise.", "---", "### Key Takeaways:", "- Always collect like terms before solving.\n- Using conjugates eliminates radicals in denominators.\n- Rational expressions simplify complex equations effectively.", "Mastering these steps makes solving radical-containing equations intuitive and efficient.", "---", "If you're solving similar equations or exploring advanced algebra, remember: truth lies beneath the surface — with careful steps, clarity follows."]