Question 150900
{{{(x-3)^(1/2) + (2x)^(1/2)= 3}}} Start with the given equation.



{{{sqrt(x-3) + sqrt(2x)= 3}}} Convert the equation from rational exponent notation to radical notation.



{{{sqrt(x-3) = 3- sqrt(2x)}}} Subtract {{{sqrt(2x)}}} from both sides.



{{{x-3 = (3- sqrt(2x))^2}}} Square both sides.



{{{x-3 = 9-6*sqrt(2x)+2x}}} FOIL



{{{x-3 -9-2x= -6*sqrt(2x)}}} Subtract 9 from both sides. Subtract {{{2x}}} from both sides.



{{{-x-12= -6*sqrt(2x)}}} Combine like terms.



{{{(-x-12)^2= 36*(2x)}}} Square both sides.



{{{(-x-12)^2= 72x}}} Multiply



{{{x^2+24x+144= 72x}}} FOIL.



{{{x^2+24x+144-72x=0}}} Subtract {{{72x}}} from both sides.



{{{x^2-48x+144=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-48}}}, and {{{c=144}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-48) +- sqrt( (-48)^2-4(1)(144) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-48}}}, and {{{c=144}}}



{{{x = (48 +- sqrt( (-48)^2-4(1)(144) ))/(2(1))}}} Negate {{{-48}}} to get {{{48}}}. 



{{{x = (48 +- sqrt( 2304-4(1)(144) ))/(2(1))}}} Square {{{-48}}} to get {{{2304}}}. 



{{{x = (48 +- sqrt( 2304-576 ))/(2(1))}}} Multiply {{{4(1)(144)}}} to get {{{576}}}



{{{x = (48 +- sqrt( 1728 ))/(2(1))}}} Subtract {{{576}}} from {{{2304}}} to get {{{1728}}}



{{{x = (48 +- sqrt( 1728 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (48 +- 24*sqrt(3))/(2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (48)/(2) +- (24*sqrt(3))/(2)}}} Break up the fraction.  



{{{x = 24 +- 12*sqrt(3)}}} Reduce.  



{{{x = 24+12*sqrt(3)}}} or {{{x = 24-12*sqrt(3)}}} Break up the expression.  



So the possible answers are {{{x = 24+12*sqrt(3)}}} or {{{x = 24-12*sqrt(3)}}} 





However, if you plug in {{{x = 24+12*sqrt(3)}}}, the equation will not be true. 



So the only answer is {{{x = 24-12*sqrt(3)}}}