Question

How do you solve the equation `x^2-2/3x-26/9=0` by completing the square?

Answer 1

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

Explanation:

Given:

`x^2-2/3x-26/9 = 0`

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

Use this with `a=(x-1/3)` and `b=sqrt(3)` as follows:

`0 = x^2-2/3x-26/9`

`color(white)(0) = x^2-2/3x+1/9-3`

`color(white)(0) = (x-1/3)^2-(sqrt(3))^2`

`color(white)(0) = ((x-1/3)-sqrt(3))((x-1/3)+sqrt(3))`

`color(white)(0) = (x-1/3-sqrt(3))(x-1/3+sqrt(3))`

Hence:

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