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`x/(x^2-2)=-1/x` Solve the equation by cross multiplying. Check for extraneous solutions.

`x/(x^2-2)=(-1)/x`


Cross multiply,


`x*x=-1(x^2-2)`


`x^2=-x^2+2`


`x^2+x^2-2=0`


`2x^2-2=0`


Factorize,


`2(x^2-1)=0`


`2(x+1)(x-1)=0`


Use the zero product property,


`x+1=0`  or  `x-1=0`


`x=-1`  or  `x=1`


Now let's check the solutions by plugging them in the original equation,


For x=-1,


`(-1)/((-1)^2-2)=(-1)/(-1)`


`(-1)/(1-2)=1`


`(-1)/(-1)=1`


`1=1`


It's true.


For x=1,


`1/(1^2-2)=(-1)/1`


`1/(1-2)=-1`


`1/(-1)=-1`


`-1=-1`


It's true.


S, the solutions are -1 and 1.

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