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Question

The value of the expression $$x^4-8x^3+18x^2-8x+2$$ when $$x=2+\sqrt 3$$


A
2
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B
1
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C
0
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D
3
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Solution

The correct option is B $$1$$
$$x=2+\sqrt { 3 } \\$$ 
or$$\quad { (x-2) }^{ 2 }={ (\sqrt { 3 } ) }^{ 2 }\\$$ 
or$$\quad { x }^{ 2 }-4x+1=0\\$$ 
or$$\quad { (x-2) }^{ 4 }=9\\$$ 
or$$\quad { x }^{ 4 }-8{ x }^{ 3 }+24{ x }^{ 2 }-32x+16=9\\$$ 
or$$\quad { x }^{ 4 }-8{ x }^{ 3 }+18{ x }^{ 2 }-8x+2+6({ x }^{ 2 }-4x+1)-1=0\\$$
As$$\quad { x }^{ 2 }-4x+1=0,\quad$$ 
we get,
$$\\ { x }^{ 4 }-8{ x }^{ 3 }+18{ x }^{ 2 }-8x+2=1$$

Mathematics

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