Question

Solve for x: 4 ( −1/2)^2 - 4(x+1/x) + 1=0

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Answer 1

Step-by-step explanation:

If x^2-4x+1=0, then what is x^9+x^7-194x^5-194x^3?

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x^9+x^7-194x^5-194x^3 =

x^7(x^2+1)-194*x^3*(x^2+1)

but x^2–4x+1= 0 → x^2+1=4x

x^7(4x)-194*x^3*(4x)=

4x(x^7–194x^3)=

4x^4(x^4–194)

but x^2–4x+1= 0 → Delta=16–4=12 →

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x=(4+(12)^(1/2))/2

→ x=(4+(4*3)^(1/2))/2

→ x=(4+2*(3)^(1/2))/2

→ x=2+(3)^(1/2)

→ x^2= 4+4*(3)^(1/2)+3= 7+ 4*(3)^(1/2)

→ x^4= 49+2*7*4*(3)^(1/2)+16*3

→ x^4= 97+56*(3)^(1/2)

→x^9+x^7-194x^5-194x^3 =4x^4(x^4–194)

= 4*(97+56*(3)^(1/2))*[97+56*(3)^(1/2)-194]

= 4*(97+56*(3)^(1/2))*[-97+56*(3)^(1/2)]

= 4*(-(97)^2+(56)^2*(3))

= 4*(-9409+3136*(3))

=-4