Question

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

Answer 1

`n=80/9`

Explanation:

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`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`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

`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`