Find the solutions of the equation...

Question

How can I solve this equation?

$$ \begin{equation*} \sqrt[3]{x-2}+\sqrt{x-1}=5 \end{equation*} $$

Frankly, I just have no idea at all!!! Thank you in advance!

Answer 1

By cubing, rearranging and squaring the given equation is reduced to a cubic equation. The original equation

\begin{equation*} \sqrt[3]{x-2}+\sqrt{x-1}=5,\qquad u=x-2,\quad v=x-1 \end{equation*}

implies that

\begin{eqnarray*} u^{1/3} &=&5-v^{1/2} \\ u &=&\left( 5-v^{1/2}\right) ^{3} \\ \left( u-125-15v\right) ^{2} &=&\left( -75v^{1/2}-v^{3/2}\right) ^{2} \\ &&\cdots \\ x^{3}-49x^{2}+2192x-18\,020 &=&0 \\ \left( x-10\right) \left( x^{2}-39x+1802\right) &=&0. \end{eqnarray*}

You still need to confirm which solutions of this last cubic equation are solutions of the original equation, because in general the equation $A^n=B^n$ has all the solutions of the equation $A=B$ but may have additional ones.