Question

can someone help?

Answer 1

Solve by completing the square:

x^2/ 25x^2−55x+10 = (5x − 10)^−1 − (5x−1)^−1

Answer 2

\[\frac{ x^2 }{ 25 x^2-55x+10 }=\left( 5x-10 \right)^{-1}-(5x-1)^{-1}\]\[R.H.S.=(5x-10)^{-1}-(5x-1)^{-1}=\frac{ 1 }{ 5x-10 }-\frac{ 1 }{ 5x-1 }\]
\[=\frac{ 5x-1-(5x-10) }{ (5x-10)(5x-1) }=\frac{ 5x-1-5x+10 }{ 25x^2-50x-5x+10 }=\frac{ 9 }{ 25x^2-50x+10 }\]
\[\frac{ x^2 }{ x^2-55x+10 }=\frac{ 9 }{ x^2-55x+10 }\]
cross multiply
\[x^2(x^2-55x+10)=9(x^2-55x+9)\]
\[x^2(x^2-55x+10)-9(x^2-55x+10)=0\]
\[(x^2-55x+10)(x^2-9)=0\]
\[x^2-55x+10 \ne 0\]
\[x^2-9=0,(x+3)(x-3)=0,x=-3,3\]