Question 151318
{{{y-2-8*sqrt(y-2)+15=0}}} Start with the given equation.



{{{y-8*sqrt(y-2)+13=0}}} Combine like terms.



{{{y-8*sqrt(y-2)=-13}}} Subtract 13 from both sides.



{{{-8*sqrt(y-2)=-13-y}}} Subtract {{{y}}} from both sides.



{{{64(y-2)=(-13-y)^2}}} Square both sides.



{{{64(y-2)=169+26y+y^2}}} FOIL



{{{64y-128=169+26y+y^2}}} Distribute



{{{0=169+26y+y^2-64y+128}}} Subtract {{{64y}}} from both sides. Add 128 to both sides.



{{{0=y^2-38y+297}}} Combine like terms



Notice we have a quadratic equation in the form of {{{ay^2+by+c}}} where {{{a=1}}}, {{{b=-38}}}, and {{{c=297}}}



Let's use the quadratic formula to solve for y



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



{{{y = (-(-38) +- sqrt( (-38)^2-4(1)(297) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-38}}}, and {{{c=297}}}



{{{y = (38 +- sqrt( (-38)^2-4(1)(297) ))/(2(1))}}} Negate {{{-38}}} to get {{{38}}}. 



{{{y = (38 +- sqrt( 1444-4(1)(297) ))/(2(1))}}} Square {{{-38}}} to get {{{1444}}}. 



{{{y = (38 +- sqrt( 1444-1188 ))/(2(1))}}} Multiply {{{4(1)(297)}}} to get {{{1188}}}



{{{y = (38 +- sqrt( 256 ))/(2(1))}}} Subtract {{{1188}}} from {{{1444}}} to get {{{256}}}



{{{y = (38 +- sqrt( 256 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{y = (38 +- 16)/(2)}}} Take the square root of {{{256}}} to get {{{16}}}. 



{{{y = (38 + 16)/(2)}}} or {{{y = (38 - 16)/(2)}}} Break up the expression. 



{{{y = (54)/(2)}}} or {{{y =  (22)/(2)}}} Combine like terms. 



{{{y = 27}}} or {{{y = 11}}} Simplify. 



So our answers are {{{y = 27}}} or {{{y = 11}}}