Question

How do you solve `(x + 1/3)^2 - 4/9 = 0`?

Answer 1

`(x+1/3)^2-4/9=0`
`implies (x+1/3)^2-(2/3)^2=0`

Use the formula named as Difference of Squares", i.e.,
`a^2-b^2=(a-b)(a+b)`

Here`a=(x+1/3)` and `b=2/3`

`implies (x+1/3-2/3)(x+1/3+2/3)=0`
`implies (x+(1-2)/3)(x+(1+2)/3)=0`
`implies (x-1/3)(x+3/3)=0`
`implies (x-1/3)(x+1)=0`

Now, use "zero product property", i.e.,
If `a.b=0,` then either `a=0 or b=0.`

Here `a=(x-1/3)` and `b=(x+1)`

`implies` either `x-1/3=0` or `x+1=0`
`implies` either `x=1/3` or `x=-1`

Solution Set`={1/3,-1}`

Answer 2

`x=-1,1/3`

Explanation:

This method does not require factorization into a difference of squares.

Add `4/9` to both sides of the equation.

`(x+1/3)^2=4/9`

Take the square root of both sides. Recall that both the positive and negative roots should be taken.

Also note that `sqrt(4/9)=sqrt4/sqrt9=2/3`.

`x+1/3=+-2/3`

Split this into two equations:

`x+1/3=2/3`

`color(red)(x=1/3`

The negative version:

`x+1/3=-2/3`

`color(red)(x=-1`