Question

X=3+√5 find value of x^2+1/x^2

Answer 1

15 + 2root5 /14 + 2root5

Hope it helps

Answer 2

Answer:

✴1st method

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x= 3 +5^1/2……………(1)

x= 3 +5^1/2……………(1)1/x= 1/(3+5^1/2) ×[(3–5^1/2)/(3–5^1/2)]

x= 3 +5^1/2……………(1)1/x= 1/(3+5^1/2) ×[(3–5^1/2)/(3–5^1/2)]1/x=(3–5^1/2)/(9–5).

x= 3 +5^1/2……………(1)1/x= 1/(3+5^1/2) ×[(3–5^1/2)/(3–5^1/2)]1/x=(3–5^1/2)/(9–5).1/x =(3–5^1/2)/4……………(2).

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➯On adding eq. (1) & (2).

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x+1/x =15/4+(3.5^1/2)/4=(3/4) (5+5^1/2)

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✰Squaring both sides

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x^2+1/x^2 +2=(9/16)(25+5+10.5^1/2)

x^2+1/x^2 +2=(9/16)(25+5+10.5^1/2)x^2+1/x^2 =(9/16) (30+10.5^1/2)-2

x^2+1/x^2 +2=(9/16)(25+5+10.5^1/2)x^2+1/x^2 =(9/16) (30+10.5^1/2)-2x^2+1/x^2=(270+90.5^1/2–32)/16

x^2+1/x^2 +2=(9/16)(25+5+10.5^1/2)x^2+1/x^2 =(9/16) (30+10.5^1/2)-2x^2+1/x^2=(270+90.5^1/2–32)/16x^2+1/x^2=(238+90.5^1/2)/16

x^2+1/x^2 +2=(9/16)(25+5+10.5^1/2)x^2+1/x^2 =(9/16) (30+10.5^1/2)-2x^2+1/x^2=(270+90.5^1/2–32)/16x^2+1/x^2=(238+90.5^1/2)/16x^2+1/x^2=(119+45.5^1/2)/8 .$<font color= </strong><strong>black</strong><strong>>$ Answer

✴2nd method:-

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x= 3+5^1/2 or squaring both sides.

x= 3+5^1/2 or squaring both sides.x^2 =9+5+6.5^1/2

x= 3+5^1/2 or squaring both sides.x^2 =9+5+6.5^1/2x^2= 14 +6.5^1/2…………………..(1).

x= 3+5^1/2 or squaring both sides.x^2 =9+5+6.5^1/2x^2= 14 +6.5^1/2…………………..(1).1/x^2= 1/(14+6.5^1/2)×[(14–6.5^1/2)/(14–6.5^1/2)].

x= 3+5^1/2 or squaring both sides.x^2 =9+5+6.5^1/2x^2= 14 +6.5^1/2…………………..(1).1/x^2= 1/(14+6.5^1/2)×[(14–6.5^1/2)/(14–6.5^1/2)].1/x^2=(14–6.5^1/2)/(196–180)

x= 3+5^1/2 or squaring both sides.x^2 =9+5+6.5^1/2x^2= 14 +6.5^1/2…………………..(1).1/x^2= 1/(14+6.5^1/2)×[(14–6.5^1/2)/(14–6.5^1/2)].1/x^2=(14–6.5^1/2)/(196–180)1/x^2=2(7–3.5^1/2)/16

x= 3+5^1/2 or squaring both sides.x^2 =9+5+6.5^1/2x^2= 14 +6.5^1/2…………………..(1).1/x^2= 1/(14+6.5^1/2)×[(14–6.5^1/2)/(14–6.5^1/2)].1/x^2=(14–6.5^1/2)/(196–180)1/x^2=2(7–3.5^1/2)/161/x^2=(7–3.5^1/2)/8………………….(2).

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✰On adding eq. (1) & (2).

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x^2+1/x^2=(14+7/8)+(6–3/8).5^1/2.

x^2+1/x^2=(14+7/8)+(6–3/8).5^1/2.x^2+1/x^2=(119+45.5^1/2)/8 . $<font color= </strong><strong>black</strong><strong>>$ Answer.