Question

When solving a radical equation, Beth and Kelly came to two different conclusions. Beth found a solution, while Kelly's solution did not work in the equation.

Create and justify two situations: one situation where Beth is correct and a separate situation where Kelly is correct.

Answer 1

\[\sqrt{x}-x=0\]
\[\sqrt{x}=x\]
\[(\sqrt{x}=x)^2\]
\[x=x^2 \rightarrow x-x^2=0 \rightarrow x(1-x)=0 \rightarrow x=0 ,1\]
both of roots are correct

Answer 2

another situation
\[\sqrt{x}+x=0\]
\[\sqrt{x}=-x\]
\[(\sqrt{x}=-x)^2\]
\[x=x^2 \rightarrow x(1-x)=0\]
the same equation
so
x=0,1
x=0 is true (check)
but x=1 is not true (check)
\[\sqrt{1}+1\neq0\]