Question

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

Answer 1

`x=80/9`

Explanation:

Given,

`sqrt(20-x)+8=sqrt(9-x)+11`

Subtract `8` from both sides.

`sqrt(20-x)+8color(white)(i)color(red)(-8)=sqrt(9-x)+11color(white)(i)color(red)(-8)`

`sqrt(20-x)=sqrt(9-x)+3`

Square both sides to get rid of the radical signs.

`(sqrt(20-x))^2=(sqrt(9-x)+3)^2`

`(sqrt(20-x))(sqrt(20-x))=(sqrt(9-x)+3)(sqrt(9-x)+3)`

Simplify.

`20-x=(9-x)+6sqrt(9-x)+9`

`20-x=9-x+6sqrt(9-x)+9`

`2=6sqrt(9-x)`

Solve for `x`.

`1/3=sqrt(9-x)`

`(1/3)^2=(sqrt(9-x))^2`

`1/9=9-x`

`x=color(green)(|bar(ul(color(white)(a/a)color(black)(80/9)color(white)(a/a)|)))`

Answer 2

`x = 80/9`

Explanation:

Regrouping and powering
`(sqrt[20 - x] - sqrt[9 - x])^2 =(11 - 8)^2`
giving
`29 - 2 sqrt[9 - x] sqrt[20 - x] - 2 x = 9`
Regrouping and powering
`(2 sqrt(9 - x) sqrt(20 - x))^2 =(29 - 9 - 2 x)^2`
giving
`9x=80`
Solving for `x` we get `x = 80/9`