Question 158211
{{{sqrt(x+2) - x = 0}}} Start with the given equation.



{{{sqrt(x+2)=x}}} Add "x" to both sides.



{{{x+2=x^2}}} Square both sides.



{{{-x^2+x+2=0}}} Subtract {{{x^2}}} from both sides.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=-1}}}, {{{b=1}}}, and {{{c=2}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(1) +- sqrt( (1)^2-4(-1)(2) ))/(2(-1))}}} Plug in  {{{a=-1}}}, {{{b=1}}}, and {{{c=2}}}



{{{x = (-1 +- sqrt( 1-4(-1)(2) ))/(2(-1))}}} Square {{{1}}} to get {{{1}}}. 



{{{x = (-1 +- sqrt( 1--8 ))/(2(-1))}}} Multiply {{{4(-1)(2)}}} to get {{{-8}}}



{{{x = (-1 +- sqrt( 1+8 ))/(2(-1))}}} Rewrite {{{sqrt(1--8)}}} as {{{sqrt(1+8)}}}



{{{x = (-1 +- sqrt( 9 ))/(2(-1))}}} Add {{{1}}} to {{{8}}} to get {{{9}}}



{{{x = (-1 +- sqrt( 9 ))/(-2)}}} Multiply {{{2}}} and {{{-1}}} to get {{{-2}}}. 



{{{x = (-1 +- 3)/(-2)}}} Take the square root of {{{9}}} to get {{{3}}}. 



{{{x = (-1 + 3)/(-2)}}} or {{{x = (-1 - 3)/(-2)}}} Break up the expression. 



{{{x = (2)/(-2)}}} or {{{x =  (-4)/(-2)}}} Combine like terms. 



{{{x = -1}}} or {{{x = 2}}} Simplify. 



So the possible answers are {{{x = -1}}} or {{{x = 2}}} 



However, it turns out that if you plug in {{{x = -1}}}, then the original equation will be false. So we must ignore {{{x = -1}}}




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Answer:



So the solution is {{{x = 2}}}