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If x=2+sqrt3, find x^(2)+(1)/(x^(2))+x+(1)/(x)

If x=2+3 x=2+\sqrt{3} , find x2+1x2+x+1x x^{2}+\frac{1}{x^{2}}+x+\frac{1}{x}

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Q. If x=2+3 x=2+\sqrt{3} , find x2+1x2+x+1x x^{2}+\frac{1}{x^{2}}+x+\frac{1}{x}
  1. Find Reciprocal and Square: Now, let's find the reciprocal of xx and then square it to get (1/x)2(1/x)^2.
    1/x=1/(2+3)1/x = 1/(2 + \sqrt{3})
    To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator.
    (1/x)⋅((2−3)/(2−3))=(2−3)/(4−(3)2)(1/x) \cdot ((2 - \sqrt{3})/(2 - \sqrt{3})) = (2 - \sqrt{3})/(4 - (\sqrt{3})^2)
    (1/x)⋅((2−3)/(2−3))=(2−3)/(4−3)(1/x) \cdot ((2 - \sqrt{3})/(2 - \sqrt{3})) = (2 - \sqrt{3})/(4 - 3)
    (1/x)=(2−3)(1/x) = (2 - \sqrt{3})
    Now square (1/x)(1/x) to get (1/x)2(1/x)^2.
    (1/x)2=(2−3)2(1/x)^2 = (2 - \sqrt{3})^2
    (1/x)2=22−2⋅2⋅3+(3)2(1/x)^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2
    (1/x)2(1/x)^200
    (\(1/x)^22 = 77 - 44\sqrt{33})
  2. Calculate x+(1/x)x + (1/x): Next, let's find the value of x+(1/x)x + (1/x).
    x+(1/x)=(2+3)+(2−3)x + (1/x) = (2 + \sqrt{3}) + (2 - \sqrt{3})
    x+(1/x)=2+3+2−3x + (1/x) = 2 + \sqrt{3} + 2 - \sqrt{3}
    x+(1/x)=4x + (1/x) = 4
  3. Add Up Terms: Now, let's add up x2x^2, (1/x)2(1/x)^2, xx, and (1/x)(1/x).
    x2+(1/x)2+x+(1/x)=(7+43)+(7−43)+4x^2 + (1/x)^2 + x + (1/x) = (7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) + 4
    x2+(1/x)2+x+(1/x)=7+43+7−43+4x^2 + (1/x)^2 + x + (1/x) = 7 + 4\sqrt{3} + 7 - 4\sqrt{3} + 4
    x2+(1/x)2+x+(1/x)=14+4x^2 + (1/x)^2 + x + (1/x) = 14 + 4
    x2+(1/x)2+x+(1/x)=18x^2 + (1/x)^2 + x + (1/x) = 18

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