r - \sqrt{5} r + 5 + \sqrt{5} = 0 \Rightarrow r(-1 - \sqrt{5}) = - (5 + \sqrt{5})

["# Solving the Quadratic Equation: ( r - \sqrt{5},r + 5 + \sqrt{5} = 0 \Rightarrow r(-1 - \sqrt{5}) = - (5 + \sqrt{5}) )", "Quadratic equations are fundamental in algebra and appear in various fields such as physics, engineering, and economics. This article provides a detailed step-by-step solution to the equation:", "[\nr - \sqrt{5},r + 5 + \sqrt{5} = 0\n]", "We’ll explain how to simplify the equation and solve for ( r ) using algebraic manipulation and proper factoring techniques. Understanding this method is essential for solving more complex equations and building a strong algebra foundation.", "## Step 1: Simplify the Equation", "Start by combining like terms on the left-hand side. The equation:", "[\nr - \sqrt{5},r + 5 + \sqrt{5} = 0\n]", "Combine the ( r )-terms:", "[\n(1 - \sqrt{5})r + (5 + \sqrt{5}) = 0\n]", "---", "## Step 2: Isolate the Variable ( r )", "To isolate ( r ), move the constant term ( (5 + \sqrt{5}) ) to the right-hand side:", "[\n(1 - \sqrt{5})r = - (5 + \sqrt{5})\n]", "This matches the target form:", "[\nr(-1 - \sqrt{5}) = - (5 + \sqrt{5})\n]", "Note: The factor ( 1 - \sqrt{5} ) was rewritten as ( - ( -1 + \sqrt{5} ) ) to highlight the negative sign—this helps in simplifying further steps.", "---", "## Step 3: Solve for ( r )", "Divide both sides by ( - (1 - \sqrt{5}) ), or equivalently, multiply both sides by ( \frac{1}{-1 - \sqrt{5}} ):", "[\nr = \frac{ - (5 + \sqrt{5}) }{ - (1 - \sqrt{5}) } = \frac{5 + \sqrt{5}}{1 - \sqrt{5}}\n]", "To simplify the fraction, rationalize the denominator by multiplying numerator and denominator by the conjugate ( 1 + \sqrt{5} ):", "[\nr = \frac{5 + \sqrt{5}}{1 - \sqrt{5}} \cdot \frac{1 + \sqrt{5}}{1 + \sqrt{5}} = \frac{(5 + \sqrt{5})(1 + \sqrt{5})}{(1 - \sqrt{5})(1 + \sqrt{5})}\n]", "---", "## Step 4: Expand the Numerator and Denominator", "First, expand the denominator using the difference of squares:", "[\n(1 - \sqrt{5})(1 + \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4\n]", "Now expand the numerator:", "[\n(5 + \sqrt{5})(1 + \sqrt{5}) = 5 \cdot 1 + 5 \cdot \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \cdot \sqrt{5} = 5 + 5\sqrt{5} + \sqrt{5} + 5 = 10 + 6\sqrt{5}\n]", "---", "## Step 5: Final Simplification", "Substitute back:", "[\nr = \frac{10 + 6\sqrt{5}}{-4} = -\frac{10 + 6\sqrt{5}}{4}\n]", "Simplify by factoring numerator:", "[\nr = -\frac{2(5 + 3\sqrt{5})}{4} = -\frac{5 + 3\sqrt{5}}{2}\n]", "---", "## Conclusion", "The solution to the quadratic equation ( r - \sqrt{5},r + 5 + \sqrt{5} = 0 ) simplifies to:", "[\nr = -\frac{5 + 3\sqrt{5}}{2}\n]", "This derivation shows how algebraic manipulation — combining like terms, isolating variables, and rationalizing denominators — enables precise solutions to radical equations. Mastering these steps empowers students and professionals to tackle advanced mathematical problems with confidence.", "---", "### Key Takeaways:", "- Combine like terms to simplify expressions.\n- Isolate the variable by moving constants appropriately.\n- Use algebraic identities (difference of squares) and rationalization for cleaner results.\n- Factoring and simplifying rational expressions ensures accurate final answers.", "Understanding equations like ( r(-1 - \sqrt{5}) = - (5 + \sqrt{5}) ) is not only key to solving quadratic problems but also foundational in higher mathematics.", "---", "Keywords: solve quadratic equation, r - √5 r + 5 + √5 = 0, r(-1 - √5) = - (5 + √5), algebraic solution, rationalizing denominator, simplifying radicals, quadratic formulas, solving equations with irrationals."]