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

["Mastering the Equation: Solving 2r + 2 = (3 + √5)r − (3 + √5)", "Solving linear equations is a fundamental skill in algebra that enables students and math enthusiasts to unlock complex problem-solving possibilities. One intriguing equation that often challenges learners demonstrates both rational and irrational coefficients:", "$$\n2r + 2 = (3 + \sqrt{5})r - (3 + \sqrt{5})\n$$", "Understanding how to solve this equation not only strengthens algebraic technique but also builds confidence in working with irrational numbers. In this article, we’ll walk through the step-by-step process of solving the equation, simplify expressions with √5, and explain key algebraic concepts—all optimized for clarity and SEO-friendly content.", "---", "### Step 1: Simplify and Rearrange Terms", "Start by bringing all terms involving ( r ) to one side and constant terms to the other:", "$$\n2r - (3 + \sqrt{5})r = - (3 + \sqrt{5}) - 2\n$$", "Simplify both sides by factoring ( r ) and combining constants:", "Left side:\n$$\n(2 - (3 + \sqrt{5}))r = (-1 - \sqrt{5})r\n$$", "Right side:\n$$\n- (3 + \sqrt{5}) - 2 = -5 - \sqrt{5}\n$$", "Now the equation is:", "$$\n(-1 - \sqrt{5})r = -5 - \sqrt{5}\n$$", "---", "### Step 2: Solve for ( r ) by Isolating the Variable", "Divide both sides by ( -1 - \sqrt{5} ). To rationalize and simplify, multiply numerator and denominator by the conjugate of the coefficient ( -1 - \sqrt{5} ), which is ( -1 + \sqrt{5} ):", "$$\nr = \frac{-5 - \sqrt{5}}{-1 - \sqrt{5}} = \frac{-(5 + \sqrt{5})}{-(1 + \sqrt{5})} = \frac{5 + \sqrt{5}}{1 + \sqrt{5}}\n$$", "---", "### Step 3: Rationalize the Denominator", "To simplify the fraction, multiply numerator and denominator by the conjugate ( 1 - \sqrt{5} ):", "$$\nr = \frac{(5 + \sqrt{5})(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})}\n$$", "Compute the denominator using the difference of squares:", "$$\n(1 + \sqrt{5})(1 - \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4\n$$", "Expand the numerator:", "$$\n(5 + \sqrt{5})(1 - \sqrt{5}) = 5 \cdot 1 - 5\sqrt{5} + \sqrt{5} \cdot 1 - \sqrt{5} \cdot \sqrt{5} = 5 - 5\sqrt{5} + \sqrt{5} - 5 = -4\sqrt{5}\n$$", "So we have:", "$$\nr = \frac{-4\sqrt{5}}{-4} = \sqrt{5}\n$$", "---", "### Final Answer", "$$\nr = \sqrt{5}\n$$", "---", "### Why This Equation Matters: Algebra Meets Irrational Numbers", "This equation beautifully illustrates the interplay between rational and irrational terms. Mastering such problems helps develop algebraic fluency, especially when simplifying expressions containing roots. The solution—( r = \sqrt{5} )—embodies how irrational numbers systematically integrate into algebraic solutions, paving the way for advanced topics like quadratic equations, trigonometry, and number theory.", "---", "### SEO-Optimized Summary & Keywords", "Top Keywords:\n- Solve linear equation\n- Rational and irrational coefficients algebra\n- Solve equation with square root\n- Algebraic simplification steps\n- Simplify expression with √5", "Meta Description:\nLearn how to solve ( 2r + 2 = (3 + \sqrt{5})r - (3 + \sqrt{5}) ) step-by-step. Discover how rational and irrational numbers work in equations, including conjugate rationalization and simplifying radicals. Perfect for high school algebra and math learners.", "Optimizing your content with semantic keywords like “solve equation with sqrt,” “irrational coefficients in linear equations,” and “rational approximation techniques” enhances discoverability. Use subheadings, bullet points, and clear definitions to improve readability and SEO performance.", "---", "### Takeaways", "- Isolate ( r ) carefully when coefficients include irrational terms.\n- Rationalize denominators by multiplying by conjugates.\n- Combine like terms and simplify radicals thoroughly.\n- This equation example helps internalize solving linear equations with irrational coefficients.", "---", "Mastering equations like ( 2r + 2 = (3 + \sqrt{5})r - (3 + \sqrt{5}) ) empowers problem-solving across academic fields—from STEM courses to advanced math. Keep practicing and exploring rational and irrational expressions to heighten your algebraic mastery.", "---", "Related Reading:\n- How to solve quadratic equations with radicals\n- Simplifying algebraic expressions involving square roots\n- Rationalizing denominators step-by-step guide", "---", "If you found this explanation helpful, share it with fellow learners and subscribe for more algebra tips and solved problems!"]