Question 186671
{{{sqrt(x+5)+sqrt(2x+3)=9}}} Start with the given equation.



{{{sqrt(x+5)=9-sqrt(2x+3)}}} Subtract {{{sqrt(2x+3)}}} from both sides.



{{{x+5=(9-sqrt(2x+3))^2}}} Square both sides



{{{x+5=81-18*sqrt(2x+3)+2x+3}}} FOIL



{{{x+5=84+2x-18*sqrt(2x+3)}}} Combine like terms.



{{{x+5-84-2x=-18*sqrt(2x+3)}}} Subtract 84 from both sides. Subtract 2x from both sides.



{{{-x-79=-18*sqrt(2x+3)}}} Combine like terms.



{{{(-x-79)^2=(-18*sqrt(2x+3))^2}}} Square both sides



{{{x^2+158x+6241=324*(2x+3)}}} FOIL and square the right side



{{{x^2+158x+6241=648x+972}}} Distribute



{{{x^2+158x+6241-648x-972=0}}} Get everything to the left side.



{{{x^2-490x+5269=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-490}}}, and {{{c=5269}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-490) +- sqrt( (-490)^2-4(1)(5269) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-490}}}, and {{{c=5269}}}



{{{x = (490 +- sqrt( (-490)^2-4(1)(5269) ))/(2(1))}}} Negate {{{-490}}} to get {{{490}}}. 



{{{x = (490 +- sqrt( 240100-4(1)(5269) ))/(2(1))}}} Square {{{-490}}} to get {{{240100}}}. 



{{{x = (490 +- sqrt( 240100-21076 ))/(2(1))}}} Multiply {{{4(1)(5269)}}} to get {{{21076}}}



{{{x = (490 +- sqrt( 219024 ))/(2(1))}}} Subtract {{{21076}}} from {{{240100}}} to get {{{219024}}}



{{{x = (490 +- sqrt( 219024 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (490 +- 468)/(2)}}} Take the square root of {{{219024}}} to get {{{468}}}. 



{{{x = (490 + 468)/(2)}}} or {{{x = (490 - 468)/(2)}}} Break up the expression. 



{{{x = (958)/(2)}}} or {{{x =  (22)/(2)}}} Combine like terms. 



{{{x = 479}}} or {{{x = 11}}} Simplify. 



So the possible answers are {{{x = 479}}} or {{{x = 11}}}




However, if you plug in {{{x = 479}}}, the original equation will NOT be true. So {{{x = 479}}} is NOT a solution




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Answer:


So the solution is {{{x = 11}}}