Question

How do you solve the equation `1/2(x-4)^2=8`?

Answer 1

`color(blue)(x=8,0`

Explanation:

Solve:

`1/2(x-4)^2=8`

Multiply both sides by `2`.

`color(red)cancel(color(black)(2))xx1/color(red)cancel(color(black)(2))(x-4)^2=8xx2`

Simplify.

`(x-4)^2=16`

Take the square root of both sides.

`sqrt((x-4)^2)=+-sqrt16`

Simplify.

`x-4=+-4`

Add `4` to both sides and solve for `x`.

`x=4+-4`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`x=4-4`

`x=0`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`x=4+4`

`x=8`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`x=8,0`

Answer 2

`x=0` and `x=8`
or
`x=0,8`

Explanation:

Order of operations.
Exponents first.
`1/2(x-4)^2=8` `--->` `1/2(x^2-8x+16)=8`

Now divide both sides by `1/2`

`cancel(1/2)(x^2-8x+16)=8-:1/2`

`x^2-8x+16=16`

Subtract 16 from both sides

`x^2-8xcancel(+16)=cancel16`
`x^2-8x=0`

Factor "x"
`x(x-8)=0`

So now we have
`x=0` and `x-8=0`

Therefore,
`x=0` and `x=8`
or
`x=0,8`