Simplify :
4+54−5+4−54+5\dfrac{4 + \sqrt5}{4 - \sqrt5} + \dfrac{4 - \sqrt5}{4 + \sqrt5}4−54+5+4+54−5
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4+54−5+4−54+5=(4+5)×(4+5)+(4−5)×(4−5)(4+5)×(4−5)=(4+5)2+(4−5)2(4)2−(5)2=16+5+85+16+5−8516−5=21+85+21−8511=4211\dfrac{4 + \sqrt5}{4 - \sqrt5} + \dfrac{4 - \sqrt5}{4 + \sqrt5}\\[1em] = \dfrac{(4 + \sqrt5) \times (4 + \sqrt5) + (4 - \sqrt5) \times (4 - \sqrt5)}{(4 + \sqrt5) \times (4 - \sqrt5)}\\[1em] = \dfrac{(4 + \sqrt5)^2 + (4 - \sqrt5)^2} {(4)^2 - (\sqrt5)^2}\\[1em] = \dfrac{16 + 5 + 8\sqrt5 + 16 + 5 - 8\sqrt5}{16 - 5}\\[1em] = \dfrac{21 + 8\sqrt5 + 21 - 8\sqrt5}{11}\\[1em] = \dfrac{42}{11}4−54+5+4+54−5=(4+5)×(4−5)(4+5)×(4+5)+(4−5)×(4−5)=(4)2−(5)2(4+5)2+(4−5)2=16−516+5+85+16+5−85=1121+85+21−85=1142
Hence, 4+54−5+4−54+5=4211\dfrac{4 + \sqrt5}{4 - \sqrt5} + \dfrac{4 - \sqrt5}{4 + \sqrt5} = \dfrac{42}{11}4−54+5+4+54−5=1142.
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