If x= (√3+1)/2, then what is the value of 4x^3 +2x^2 - 8x + 7

Please support your answer with an explanation

x = (鈭? + 1)/2

x鲁 = (鈭? + 1)鲁/2鲁

x鲁 = (鈭? + 1)虏(鈭? + 1)/8

x鲁 = (3 + 2鈭? + 1)(鈭? + 1)/8

x鲁 = (4 + 2鈭?)(鈭? + 1)/8

x鲁 = (4鈭? + 4 + 6 + 2鈭?)/8

x鲁 = (10 + 6鈭?)/8

x鲁 = (5 + 3鈭?)/4

4x鲁 = 5 + 3鈭? 鈫?memorize this result


x = (鈭? + 1)/2

x虏 = (鈭? + 1)虏/2虏

x虏 = (3 + 2鈭? + 1)/4

x虏 = (4 + 2鈭?)/4

x虏 = (2 + 鈭?)/2

2x虏 = 2 + 鈭? 鈫?memorize this result


x = (鈭? + 1)/2

8x = 8(鈭? + 1)/2

8x = 4(鈭? + 1)

8x = 4鈭? + 4 鈫?memorize this result




= 4x鲁 + 2x虏 - 8x + 7

= (5 + 3鈭?) + (2 + 鈭?) - (4鈭? + 4) + 7

= 5 + 3鈭? + 2 + 鈭? - 4鈭? - 4 + 7

= (5 + 7 + 2 - 4) + (3鈭? + 鈭? - 4鈭?)

= 10

i) 4x鲁 + 2x虏 - 8x + 7 = (2x虏)(2x + 1) - 8x + 7

ii) Given: x = (鈭? + 1)/2; ==> 2x = (鈭? + 1)
==> 2x虏 = (2x)*(x) = (鈭? + 1)*(鈭? + 1))/2 = (3 + 1 + 2鈭?)/2 = (鈭? + 2)

and (2x + 1) = 2(鈭? + 1)/2 + 1 = 鈭? + 2

iii) Substituting all the above in step (i):
4x鲁 + 2x虏 - 8x + 7 = (2x虏)(2x + 1) - 8x + 7 = (鈭? + 2)*(鈭? + 2) - 8((鈭? + 1)/2 + 7
= 3 + 4 + 4鈭? - 4鈭? - 4 + 7 = 10

Thus value of 4x鲁 + 2x虏 - 8x + 7 = 10

If x = (鈭?+1)/2, then 4x鲁 = 4((鈭?+1)鲁/8)
= 4(6鈭? + 10)/8
= (1/2)(6鈭? + 10)
= 3鈭? + 5.

2x虏 = 2((鈭?+1)/2)虏
= 2((鈭?+1)虏/4)
= 2(4 + 2鈭?)/4
= (1/2)(4 + 2鈭?)
= 2 + 鈭?.

8x = 8(鈭?+1)/2
= 4(鈭? + 1)
= 4鈭? + 4.

So, 4x鲁 + 2x虏 - 8x + 7
= 4((鈭?+1)/2)鲁 + 2((鈭?+1)/2)虏 - 8(鈭?+1)/2 + 7
= (3鈭? + 5) + (2 + 鈭?) - (4鈭? + 4) + 7
= 10.