Solve :-

x^3 - 7x^2 +13x + 17 when x = 2 + sqrt(3)

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x^3 -7x^2 +13x+17

Substitute with x= 2+sqrt(3)

=(2+sqrt(3))^3 -7(2+sqrt3)^2 +13(2+sqrt3)+17

= (2+sqrt(3)(7+4sqrt3) -7(7+4sqrt3)+13(2+sqrt3)+17

= 26+ 15sqrt3 -49-28sqrt3 + 26 + 13sqrt3+17

= 20 +28sqrt3 -28sqrt3

= 20

To solve this expression, we'll substitute x by the given value 2 + sqrt(3).

We'll get:

[2 + sqrt(3)]^3 -7[2 + sqrt(3)]^2 + 13[2 + sqrt(3)] + 17 =

= [2 + sqrt(3)]^2}*[2 + sqrt(3) - 7] + 26 + 13sqrt(3) + 17 =

= (4 + 4sqrt3 + 3)(-5 + sqrt3) + 13sqrt3 + 43 =

= (7 + 4sqrt3)(-5 + sqrt3) + 43 + 13sqrt3=

= -35 + 7sqrt3 - 20sqrt3 + 12 + 43 + 13sqrt3 =

= 55 - 35 + 13sqrt3 - 13sqrt3 = 20

To solve f(x) = x^3-7x^2+13x+17 , when x=2+sqr3.

Solution:

f(x+sqrt3) = (x+sqrt3)^3 -7(2+sqrt3)+13(2+sqrt3)+17

= {2^3 +3*2^2*sqrt3+3*2*(sqrt3)^2+(sqrt3)^3} -7{2^2+2*2sqrt3+3} + 13(2+sqrt3}+17

= {8+12sqrt3+18+3sqr3} - {28+28sqrt3+3}+{26+13sqrt3}+17

= {26+15sqrt3}-{49+28sqrt3}+{26+13sqrt3}+17

={26-49+26+17} + {15-28 +13}sqrt3

= 20

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