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\(\cfrac{a}{ax-1}+\cfrac{b}{bx-1}=a+b\)
or, \(\cfrac{a}{ax-1}-b+\cfrac{b}{bx-1}-a=0\)
or, \(\cfrac{a-abx+b}{ax-1}+\cfrac{b-abx+a}{bx-1}=0\)
or, \((a+b-abx)\Big[\cfrac{1}{ax-1}+\cfrac{1}{bx-1}\Big]=0\)
or, \((a+b-abx)\Big[\cfrac{bx-1+ax-1}{(ax-1)(bx-1)}\Big]=0\)
or, \((a+b-abx)\Big[\cfrac{bx+ax-2}{(ax-1)(bx-1)}\Big]=0\)

Either, \((a+b-abx)=0\)
Or, \(\cfrac{bx+ax-2}{(ax-1)(bx-1)}=0\)

When, \((a+b-abx)=0\)
Then, \( -abx=-a-b\)
or, \(x=\cfrac{a+b}{ab}\)
When, \(\cfrac{bx+ax-2}{(ax-1)(bx-1)}=0\)
Then, \( bx+ax-2=0\)
or, \(x=\cfrac{2}{a+b}\)

\(\therefore\) The roots of the quadratic equation are \(\cfrac{a+b}{ab}\) and \(\cfrac{2}{a+b}\).