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# If x = sqrt(2) - 1, then the value of x^4+4x^3+6x^2+4x+7 is

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Manager
Joined: 18 Jul 2019
Posts: 54
If x = sqrt(2) - 1, then the value of x^4+4x^3+6x^2+4x+7 is  [#permalink]

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25 Nov 2019, 01:30
1
4
00:00

Difficulty:

55% (hard)

Question Stats:

50% (02:42) correct 50% (03:52) wrong based on 12 sessions

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If $$x = \sqrt{2} - 1$$, then the value of $$x^4+4x^3+6x^2+4x+7$$ is

(A) $$(\sqrt{2} -1)^4$$
(B) $$4 + 4\sqrt{2}$$
(C) 6
(D) 10
(E) 12
VP
Joined: 19 Oct 2018
Posts: 1151
Location: India
If x = sqrt(2) - 1, then the value of x^4+4x^3+6x^2+4x+7 is  [#permalink]

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25 Nov 2019, 03:18
1
$$x+1=\sqrt{2}$$

Squaring both sides

$$x^2+2x+1=2$$

$$x^2+2x=1$$........(1)

x(x+2)=1

Squaring both sides

$$x^2(x^2+4x+4)=1$$

$$x^4+4x^3+4x^2=1$$........(2)

$$x^4+4x^3+6x^2+4x+7$$

$$=x^4+4x^3+4x^2+2x^2+4x+7$$

$$=x^4+4x^3+4x^2+2(x^2+2x)+7$$

$$= 1+2(1)+7=10$$

CaptainLevi wrote:
If $$x = \sqrt{2} - 1$$, then the value of $$x^4+4x^3+6x^2+4x+7$$ is

(A) $$(\sqrt{2} -1)^4$$
(B) $$4 + 4\sqrt{2}$$
(C) 6
(D) 10
(E) 12
If x = sqrt(2) - 1, then the value of x^4+4x^3+6x^2+4x+7 is   [#permalink] 25 Nov 2019, 03:18
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