Question

How do you solve `(x-9)^2=12`?

Answer 1

`x=-2sqrt3+9, x=-2sqrt3+9`

Explanation:

You could expand the left side, move everything to the left, and apply the quadratic formula; however, taking the root of both sides is far faster:

`sqrt((x-9)^2)=+-sqrt(12)`

Recall that `sqrt(a^2)=a`, and that `sqrt12=sqrt(4*3)=sqrt4sqrt3=2sqrt3` :

`x-9=+-2sqrt3`

`x=-2sqrt3+9, x=-2sqrt3+9`

You would have the same result even with applying the quadratic formula -- this method is just faster and cleaner.

Answer 2

`x=2sqrt3+9,` `-2sqrt3+9`

Explanation:

Solve:

`(x-9)^2=12`

Take the square root of both sides.

`x-9=+-sqrt12`

Prime factorize `12`.

`x-9=+-sqrt(2xx2xx3)=`

`x-9=+-sqrt(2^2xx3)`

Apply rule: `sqrt(a^2)=a`

`x-9=+-2sqrt3`

Add `9` to both sides.

`x=+-2sqrt3+9`

Solutions for `x`.

`x=2sqrt3+9,` `-2sqrt3+9`