How do you solve #sqrt(20-n)+8=sqrt(9-n)+11#?

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Answer 1 · 2016-10-16T11:50:45

#n=80/9#

Given:#" "sqrt(20-n)+8=sqrt(9-n)+11#

#color(brown)("When ever you have roots try and get rid of them. This may not always work")#

Subtract 8 from both sides isolating one of the roots.

#" "sqrt(20-n)=sqrt(9-n)+3#

Square both sides

#20cancel(-n)=[9cancel(-n)] + 6sqrt(9-n)+9#

Subtract 18 from both sides (9+9=18)

#2=6sqrt(9-n)#

Divide both sides by 6

#1/3=sqrt(9-n)#

Square both sides

#1/9=9-n#

#n=80/9#
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#color(blue)("Check")# comparing left to right

#sqrt(20-n)+8->sqrt(9-n)+11#

#sqrt(20-80/9)+8->sqrt(9-80/9)+11#

#sqrt(100/9)+8->sqrt(1/9)+11#

#10/3+8->1/3+11#

#34/3->34/3# Thus LHS = RHS #->color(red)(" True")#

Answer 2 · 2016-12-07T10:04:52

#sqrt(20-n)+8=sqrt(9-n)+11#

#=>sqrt(20-n)-sqrt(9-n)=11-8#

#=>sqrt(20-n)-sqrt(9-n)=3.... .[1]#

#=>1/(sqrt(20-n)-sqrt(9-n))=1/3#

#=>(sqrt(20-n)+sqrt(9-n))/(20-n-9+n)=1/3#

#=>(sqrt(20-n)+sqrt(9-n))/11=1/3#

#=>(sqrt(20-n)+sqrt(9-n))= 11/3......[2]#

Adding [1] and [2] we get

#2sqrt(20-n)=3+11/3=20/3#

#=>sqrt(20-n)=10/3#

#=>(20-n)=100/9#

#=>n=20-100/9=80/9#