Question 174721
Is the equation {{{x=sqrt(x)+2}}} ???



{{{x=sqrt(x)+2}}} Start with the given equation.



{{{x-2=sqrt(x)}}} Subtract 2 from both sides.



{{{(x-2)^2=x}}} Square both sides.



{{{x^2-4x+4=x}}} FOIL.



{{{x^2-4x+4-x=0}}} Subtract x from both sides.



{{{x^2-5x+4=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-5}}}, and {{{c=4}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-5) +- sqrt( (-5)^2-4(1)(4) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-5}}}, and {{{c=4}}}



{{{x = (5 +- sqrt( (-5)^2-4(1)(4) ))/(2(1))}}} Negate {{{-5}}} to get {{{5}}}. 



{{{x = (5 +- sqrt( 25-4(1)(4) ))/(2(1))}}} Square {{{-5}}} to get {{{25}}}. 



{{{x = (5 +- sqrt( 25-16 ))/(2(1))}}} Multiply {{{4(1)(4)}}} to get {{{16}}}



{{{x = (5 +- sqrt( 9 ))/(2(1))}}} Subtract {{{16}}} from {{{25}}} to get {{{9}}}



{{{x = (5 +- sqrt( 9 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (5 +- 3)/(2)}}} Take the square root of {{{9}}} to get {{{3}}}. 



{{{x = (5 + 3)/(2)}}} or {{{x = (5 - 3)/(2)}}} Break up the expression. 



{{{x = (8)/(2)}}} or {{{x =  (2)/(2)}}} Combine like terms. 



{{{x = 4}}} or {{{x = 1}}} Simplify. 



Take note that if we plug in {{{x = 1}}}, we get an inconsistent equation. So {{{x = 1}}} is NOT a solution





So the answer is {{{x = 4}}}