How do I find the extraneous solution of sqrt(x-3)-sqrt(x)=3?

1 Answer
Sep 28, 2014

sqrt{x-3}-sqrt{x}=3

by subtracting sqrt{x-3},

Rightarrow -sqrt{x}=3-sqrt{x-3}

by squaring both sides,

Rightarrow x=9-6sqrt{x-3}+x-3

by cancelling out x's,

Rightarrow0=6-6sqrt{x-3}

by dividing by 6,

Rightarrow 0=1-sqrt{x-3}

by adding sqrt{x-3},

Rightarrow sqrt{x-3}=1

by squaring both sides,

Rightarrow x-3=1

by adding 3,

Rightarrow x=4

Even though all implications above are valid, some of them are not reversible. As a result, x=4 we found does not satisfy the original equation, which makes it extraneous.