$$(x+ \frac{1}{x} )+( x^{2} + \frac{1}{ x^{2} } )+( x^{3} + \frac{1}{ x^{3} } )=28 Suppose x+ \frac{1}{x}=a \Longrightarrow x^{2} + \frac{1}{ x^{2} } = a^{2} -2 \Longrightarrow x^{3} + \frac{1}{ x^{3} } = a^{3} -3a \Longrightarrow a+ a^{2} -2+ a^{3} -3a=28 \Longrightarrow a^{3} + a^{2} -2a-30=0 \Longrightarrow ( a^{3} -27)+( a^{2} -2a-3)=0 \Longrightarrow (a-3)( a^{2} +3a+9)+(a-3)(a+1)=0 \Longrightarrow (a-3)\underbrace{ (a^{2} +4a+10)} > 0 =0 \Longleftrightarrow a=3 \Longrightarrow x+ \frac{1}{x} =3 \Longrightarrow x^{2} -3x+1=0 \Longrightarrow x= \frac{3 \pm \sqrt{5} }{2} \Longrightarrow 2x-3= \pm \sqrt{5} \Longleftrightarrow (2x-3)^{2} =5 $$
