If \(\dfrac{1}{x+k}+\dfrac{1}{x+2k}+\dfrac{1}{x+5k}=\dfrac{1}{k}\), then which one of the following is the solution of this equation?
\(k\)
Substitute \(x = k\): \(\dfrac{1}{2k}+\dfrac{1}{3k}+\dfrac{1}{6k} = \dfrac{3+2+1}{6k} = \dfrac{6}{6k} = \dfrac{1}{k}\), which satisfies the equation. (Setting \(x=tk\) and simplifying gives \(t^3+5t^2+t-7=0\), i.e. \((t-1)(t^2+6t+7)=0\); the other two roots are irrational and negative, so \(t=1\), i.e. \(x=k\), is the matching solution.)
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