Expression with powers

If x-1/x=5, find the value of x⁴+1/x⁴

Correct answer:

v =  727

Step-by-step explanation:

$$\begin{gathered} x-1\mathrm{/}x=5 \\ {x}^2-1=5x \\ {x}^2-5x-1=0 \\ a=1;b=-5;c=-1 \\ D=b^2-4ac=5^2-4\cdot 1\cdot (-1)=29 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{5\pm \sqrt{29}}{2}} \\ {x}_{1,2}=2.5\pm 2.6925824035673 \\ {x}_1=5.1925824035673 \\ {x}_2=-0.19258240356725 \\ \text{ Factored form of the equation: } \\ (x-5.1925824035673)(x+0.19258240356725)=0 \\ {v}_1=x_1^4+1\mathrm{/}{x}_1^4=5.1926^4+1\mathrm{/}5.1926^4=727 \\ {v}_2=x_2^4+1\mathrm{/}{x}_2^4=(-0.1926{)}^4+1\mathrm{/}(-0.1926{)}^4=727 \\ {v}_1=v_2 \\ v=v_1=727 \end{gathered}$$

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