\( rac{-\sqrt{5}+1}{-\sqrt{5}-1} = rac{\sqrt{5}-1}{\sqrt{5}+1} = rac{(\sqrt{5}-1)^2}{5 - 1} = rac{6 - 2\sqrt{5}}{4} \), but we earlier derived it correctly as \( rac{3 - \sqrt{5}}{2} \), so it checks.

["Simplifying the Radical Expression: Proving ( \dfrac{-\sqrt{5}+1}{-\sqrt{5}-1} = \dfrac{\sqrt{5}-1}{\sqrt{5}+1} = \dfrac{(\sqrt{5}-1)^2}{5 - 1} = \dfrac{6 - 2\sqrt{5}}{4} ) Correctly", "When working with irrational numbers like ( \sqrt{5} ), rationalizing expressions and simplifying complex fractions can be both challenging and rewarding. One often-encountered fraction involving square roots is:", "[\n\dfrac{-\sqrt{5} + 1}{-\sqrt{5} - 1}\n]", "This expression may seem daunting at first, but through strategic algebraic manipulation and rationalization, we can simplify it neatly. Below, we walk step-by-step through the correct derivation, confirming that the expression simplifies correctly and checking its value in a clear, educational way.", "---", "### Step 1: Rewrite the Original Fraction", "Start with the original form:", "[\n\dfrac{-\sqrt{5} + 1}{-\sqrt{5} - 1}\n]", "To simplify, we first factor signs in numerator and denominator to improve clarity:", "[\n= \dfrac{1 - \sqrt{5}}{-(\sqrt{5} + 1)} = -\dfrac{1 - \sqrt{5}}{\sqrt{5} + 1}\n]", "Alternatively, multiply numerator and denominator by (-1) to eliminate the negative in the denominator:", "[\n= \dfrac{-\sqrt{5} + 1}{-\sqrt{5} - 1} = \dfrac{(-\sqrt{5} + 1)}{-(\sqrt{5} + 1)} = \dfrac{1 - \sqrt{5}}{-(\sqrt{5} + 1)} = \dfrac{-(1 - \sqrt{5})}{\sqrt{5} + 1} = -\dfrac{1 - \sqrt{5}}{\sqrt{5} + 1}\n]", "But an easier path arises when we observe symmetry and directly transform the fraction.", "---", "### Step 2: Multiply Numerator and Denominator to Rationalize Form", "A powerful method is to multiply numerator and denominator by the conjugate of the denominator to rationalize the expression.", "Denominator: ( -\sqrt{5} - 1 ) → conjugate: ( -\sqrt{5} + 1 )", "But note: multiplying numerator and denominator by the conjugate of the denominator helps eliminate radicals.", "Let us multiply numerator and denominator by:", "[\n(-\sqrt{5} + 1)\n]", "But wait — a smarter approach: recognize that both forms are equivalent, and we can simplify algebraically:", "Let\n[\nx = \dfrac{-\sqrt{5} + 1}{-\sqrt{5} - 1}\n]", "Multiply numerator and denominator by the conjugate of the denominator, which is ( -\sqrt{5} + 1 ), but actually, to eliminate radicals cleanly, multiply by the conjugate of the denominator as is:", "[\nx = \dfrac{1 - \sqrt{5}}{-1 - \sqrt{5}} \quad \ ext{(rewriting numerator and denominator)}\n]", "But to rationalize efficiently, multiply numerator and denominator by:", "[\n(-\sqrt{5} + 1) \quad \ ext{(the conjugate of the denominator)}\n]", "Actually, note:\nDenominator: ( a + b = -\sqrt{5} - 1 ), conjugate is ( -\sqrt{5} + 1 )", "So:", "[\nx = \dfrac{1 - \sqrt{5}}{-\sqrt{5} - 1} \cdot \dfrac{-\sqrt{5} + 1}{-\sqrt{5} + 1} = \dfrac{(1 - \sqrt{5})(-\sqrt{5} + 1)}{(-\sqrt{5} - 1)(-\sqrt{5} + 1)}\n]", "Let’s compute numerator and denominator separately.", "---", "### Step 3: Compute Numerator and Denominator", "Numerator:\n[\n(1 - \sqrt{5})(-\sqrt{5} + 1) = (1)(-\sqrt{5}) + (1)(1) + (-\sqrt{5})(-\sqrt{5}) + (-\sqrt{5})(1)\n]", "[\n= -\sqrt{5} + 1 + 5 - \sqrt{5} = (-\sqrt{5} - \sqrt{5}) + (1 + 5) = -2\sqrt{5} + 6\n]", "So, numerator = ( 6 - 2\sqrt{5} )", "Denominator:\n[\n(-\sqrt{5} - 1)(-\sqrt{5} + 1)\n]", "This is of the form ( (a - b)(a + b) = a^2 - b^2 ), with ( a = -\sqrt{5}, b = 1 )", "[\n= (-\sqrt{5})^2 - (1)^2 = 5 - 1 = 4\n]", "So, denominator = 4", "Therefore:", "[\nx = \dfrac{6 - 2\sqrt{5}}{4} = \dfrac{2(3 - \sqrt{5})}{4} = \dfrac{3 - \sqrt{5}}{2}\n]", "This confirms the correct simplification.", "---", "### Step 4: Verification via Alternative Path — Confirming Equivalence", "Earlier, a secondary expression was given:", "[\n\dfrac{-\sqrt{5} + 1}{-\sqrt{5} - 1} = \dfrac{\sqrt{5} - 1}{\sqrt{5} + 1}\n]", "Let’s verify this equivalence algebraically.", "Multiply numerator and denominator of the left-hand side by (-1):", "[\n\dfrac{-\sqrt{5} + 1}{-\sqrt{5} - 1} = \dfrac{ -(\sqrt{5} - 1) }{ -(\sqrt{5} + 1) } = \dfrac{\sqrt{5} - 1}{\sqrt{5} + 1}\n]", "✅ This confirms the equivalence.", "Now, simplifying ( \dfrac{\sqrt{5} - 1}{\sqrt{5} + 1} ) by rationalizing:", "Multiply numerator and denominator by ( \sqrt{5} - 1 ):", "[\n\dfrac{(\sqrt{5} - 1)^2}{(\sqrt{5} + 1)(\sqrt{5} - 1)} = \dfrac{(\sqrt{5} - 1)^2}{5 - 1} = \dfrac{5 - 2\sqrt{5} + 1}{4} = \dfrac{6 - 2\sqrt{5}}{4} = \dfrac{3 - \sqrt{5}}{2}\n]", "✅ Matches our earlier result.", "---", "### Why This Simplification Matters", "Expressing such radical expressions in simplest form is essential in algebra, trigonometry, and complex number arithmetic. These techniques—rationalizing, identifying conjugates, and simplifying square differences—form foundational skills.", "The value ( \dfrac{3 - \sqrt{5}}{2} \approx \dfrac{3 - 2.236}{2} = \dfrac{0.764}{2} \approx 0.382 ), and numerically checking:", "Original:\n[\n\dfrac{-\sqrt{5}+1}{-\sqrt{5}-1} \approx \dfrac{-2.236 + 1}{-2.236 - 1} = \dfrac{-1.236}{-3.236} \approx 0.382\n]", "Matches.", "---", "### Conclusion", "The expression ( \dfrac{-\sqrt{5} + 1}{-\sqrt{5} - 1} ) simplifies rigorously to ( \dfrac{\sqrt{5} - 1}{\sqrt{5} + 1} ), then through rationalization and algebraic manipulations to ( \dfrac{3 - \sqrt{5}}{2} ), with all steps verified step-by-step. This not only confirms correctness but builds deeper understanding of irrational number manipulation.", "For learners and practitioners, mastering such identities enables fluency in advanced algebra and supports confident problem-solving in higher mathematics.", "---", "#### Key Takeaways:\n- Use conjugates to rationalize denominators.\n- Equivalent forms can be verified via algebraic multiplication.\n- Simplification using identities ensures clarity and correctness.\n- Confirming via decimal approximation adds interpretative value.", "Keywords: ( \dfrac{-\sqrt{5}+1}{-\sqrt{5}-1} ), rationalizing, simplifying radicals, ( \dfrac{3 - \sqrt{5}}{2} ), algebraic verification, conjugate method, simplified expression."]