Question

How do you solve `x^2-6x-9=0` by completing the square?

Answer 1

`(x-3)^2-18`

Explanation:

`x^2-6x-9 = (x^2-6x+9)-18`

`x^2-6x+9 = (x-3)(x-3) = (x-3)^2`

`(x-3)^2-18 = (x^2-6x+9)-(9+9) = x^2-6x+9-9-9`

`= x^2-6x-9`

Answer 2

`x = 3+-3sqrt(2)`

Explanation:

The difference of squares identity can be written:

`A^2-B^2 = (A-B)(A+B)`

Use this with `A=(x-3)` and `B=3sqrt(2)` as follows:

`0 = x^2-6x-9`

`color(white)(0) = x^2-2(x)(3)+3^2-18`

`color(white)(0) = (x-3)^2-(3sqrt(2))^2`

`color(white)(0) = ((x-3)-3sqrt(2))((x-3)+3sqrt(2))`

`color(white)(0) = (x-3-3sqrt(2))(x-3+3sqrt(2))`

Hence:

`x = 3+-3sqrt(2)`