Question

How do you solve using the completing the square method `9x^2-12x+5=0`?

Answer 1

I found:
`x_1=(2+i)/3`
`x_2=(2-i)/3`

Explanation:

Let us manipulate a bit our expression (take the `5` to the right):
`9x^2-12x=-5`
let us add and subtract `4` to the left:
`9x^2-12xcolor(red)(+4-4)=-5`
rearrange:
`9x^2-12x+4=4-5`
let us recognize the square on the left as:
`color(blue)((3x-2)^2)=-1`
BUT
if we try to solve taking the root of both sides we will get a negative square root!
I am not sure you know about them but we can write `sqrt(-1)=i` (the immaginarty unit) and keep on going solving our equation as:
`sqrt((3x-2)^2)=sqrt(-1)`
`3x-2=+-i`
so we have 2 solutions:
`x_1=(2+i)/3`
`x_2=(2-i)/3`