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Solve the equation. Write all proposed solutions. Cross out those that are extraneous.

x+2+2=x\sqrt { - x + 2 } + 2 = x


Answered 1 year ago
Answered 1 year ago
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Evaluating the given radical equation, we,

x+2=x2Add 2 both sides(x+2)2=(x2)2Square both sidesx+2=x24x+4Expand the perfect square0=x23x+2Add x and 2 both sides0=x2x2x+2Rewrite the middle term0=x(x1)2(x1)0=(x2)(x1)Factorx2=0    or    x1=0Set each factor equal to 0x=2    or    x=1\begin{aligned} \sqrt{-x+2}&=x-2&&\texttt{Add $-2$ both sides}\\(\sqrt{-x+2})^{2}&=(x-2)^{2}&&\texttt{Square both sides}\\-x+2&=x^{2}-4x+4&&\texttt{Expand the perfect square}\\0&=x^{2}-3x+2&&\texttt{Add $x$ and $-2$ both sides}\\0&=x^{2}-x-2x+2&&\texttt{Rewrite the middle term}\\0&=x(x-1)-2(x-1)\\0&=(x-2)(x-1)&&\texttt{Factor}\\x-2=0~~~~ &\texttt{or}~~~~x-1=0&&\texttt{Set each factor equal to 0}\\x=2~~~~&\texttt{or}~~~~x=1 \end{aligned}

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