Question

How to do factorization? i.e. 6x^2 - 8x + 4 = 0 ?
And to do the 'complete the square' on it as well?

Answer 1

So to do this one, we will divide both sides of the equation by 6 to get the first term to just be x^2. You'll want to do this most of the time, since it'll make your life easier.

See, what we're going to do is try to make it fit into a simple squared formula like this:

(x+m)^2

So if you ever have trouble figuring it out, just write that, and then multiply the two to get:

x^2+2mx+m^2=(x+m)^2

Alright, now if you look at what you have, you'll have something multiplied by x. Make that equal to 2m. Now you'll be able to solve for m and square it, then add it to both sides of the formula. So for example if you have:

x^2+14x-6=0

you can see that 14 is where the 2m is on the other thing. So,

14=2m
m=7
m^2=49

So what you can do to make your life a little easier is add 6 to both sides so it looks like this, and we'll use that m^2 in a second.

x^2+14x-6=0
add 6 to both sides
x^2+14x=6

now if you add 49 (m^2) to both sides you get:

x^2+14x+49=55

We chose m^2 very carefully so that now we can go from:

x^2+2m+m^2
to
(x+m)^2

since they're equal, right?

x^2+14x+49=55
Now we do the step
(x+7)^2=55

So take the square root of both sides
\[x+7=\pm \sqrt{55}\]\[x=-7\pm \sqrt{55}\]
So those are our two possible answers.

The reason for the + and - sign on the square root is because if you take any number, like -2 or 2 and square them, they both equal 4, so there are two possible choices.

That's how you complete the square basically.