If x=5+root5, find the value of 10x +7x2 - x3

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Solution

Dear student,

$A s s u m i n g t h e q u e s t i o n i s " x = 5 + \sqrt{5} . f i n d t h e v a l u e o f I f x = 5 + r o o t 5, f i n d t h e v a l u e o f 10 x - 7 x^{2} + x^{3} " W e h a v e, 10 x - 7 x^{2} + x^{3} = x [10 - 7 x + x^{2}] = x [x^{2} - 7 x + 10] = x [x^{2} - 5 x - 2 x + 10] = x [x (x - 5) - 2 (x - 5)] = x (x - 5) (x - 2) P u t x = 5 + \sqrt{5} w e g e t, = (5 + \sqrt{5}) \times (5 + \sqrt{5} - 5) \times (5 + \sqrt{5} - 2) = \underset{⏟}{(5 + \sqrt{5}) \times \sqrt{5}} \times (3 + \sqrt{5}) = (5 \sqrt{5} + 5) \times (3 + \sqrt{5}) = 15 \sqrt{5} + 15 + 25 + 5 \sqrt{5} = 20 \sqrt{5} + 40 = 20 (\sqrt{5} + 2)$
Regards

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