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If-x-2-3x-1-0-then-find-the-value-of-x-2-x-1-x-1-x-2-2-

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Question Number 191552 by MATHEMATICSAM last updated on 25/Apr/23
If x^2 − 3x + 1 = 0 then find the value of (x^2 + x + (1/x) + (1/x^2 ))^2
Ifx2−3x+1=0thenfindthevalueof(x2+x+1x+1x2)2
Answered by mr W last updated on 25/Apr/23
x^2 −3x+1=0 ⇒x+(1/x)=3 (x^2 +x+(1/x)+(1/x^2 ))^2 =(x+(1/x)+x^2 +(1/x^2 ))^2 =[(x+(1/x))+(x+(1/x))^2 −2]^2 =(3+3^2 −2)^2 =100 ✓
x2−3x+1=0⇒x+1x=3(x2+x+1x+1x2)2=(x+1x+x2+1x2)2=[(x+1x)+(x+1x)2−2]2=(3+32−2)2=100✓
Answered by Rasheed.Sindhi last updated on 26/Apr/23
AnOther Way... x^2 − 3x + 1 = 0⇒1=3x−x^2 =x(3−x) (x^2 + x + (1/x) + (1/x^2 ))^2 =(x^2 + x + ((x(3−x))/x) + ((x(3−x))/x^2 ))^2 =(x^2 + x + 3−x + ((3−x)/x))^2 =(x^2 + 3 + ((3−x)/x))^2 =(((x^3 +3x+3−x)/x))^2 =(((x^3 +2x+3×1)/x))^2 =(((x^3 +2x+3x(3−x))/x))^2 =(((x^3 +2x+9x−3x^2 )/x))^2 =(x^2 −3x+1+10)^2 =(0+10)^2 =100 ✓
AnOtherWay…x2−3x+1=0⇒1=3x−x2=x(3−x)(x2+x+1x+1x2)2=(x2+x+x(3−x)x+x(3−x)x2)2=(x2+x+3−x+3−xx)2=(x2+3+3−xx)2=(x3+3x+3−xx)2=(x3+2x+3×1x)2=(x3+2x+3x(3−x)x)2=(x3+2x+9x−3x2x)2=(x2−3x+1+10)2=(0+10)2=100✓

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