Expression with powers

If x-1/x=5, find the value of x4+1/x4

Result

v =  727

Solution:

x−1/x=5 x2−1=5x  x2−5x−1=0  a=1;b=−5;c=−1 D=b2−4ac=52−4⋅1⋅(−1)=29 D>0  x1,2=−b±D2a=5±292 x1,2=2.5±2.69258240357 x1=5.19258240357 x2=−0.192582403567   Factored form of the equation:  (x−5.19258240357)(x+0.192582403567)=0  v1=x14+1/x14=5.19264+1/5.19264=727 v2=x24+1/x24=(−0.1926)4+1/(−0.1926)4=727  v1=v2 v=v1=727x-1/x=5 \ \\ x^2-1=5x \ \\ \ \\ x^2 -5x -1=0 \ \\ \ \\ a=1; b=-5; c=-1 \ \\ D=b^2 - 4ac=5^2 - 4\cdot 1 \cdot (-1)=29 \ \\ D>0 \ \\ \ \\ x_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 5 \pm \sqrt{ 29 } }{ 2 } \ \\ x_{1,2}=2.5 \pm 2.69258240357 \ \\ x_{1}=5.19258240357 \ \\ x_{2}=-0.192582403567 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (x -5.19258240357) (x +0.192582403567)=0 \ \\ \ \\ v_{1}=x_{1}^4+1/x_{1}^4=5.1926^4+1/5.1926^4=727 \ \\ v_{2}=x_{2}^4+1/x_{2}^4=(-0.1926)^4+1/(-0.1926)^4=727 \ \\ \ \\ v_{1}=v_{2} \ \\ v=v_{1}=727

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