Question

What is the factorization of `x^2+6x+9`?

Answer 1

The factored version is `(x+3)^2`

Explanation:

Here's how I approached it: I can see that `x` is in the first two terms of the quadratic, so when I factor it down it looks like:

`(x+a)(x+b)`

And when that gets expanded it looks like:

`x^2+(a+b)x+ab`

I then looked at the system of equations:

`a+b=6`

`ab=9`

What caught my eye was that both 6 and 9 are multiples of 3. If you replace `a` or `b` with 3, you get the following (I replaced `a` for this):

`3+b=6 rArr b=3`
`3b=6 rArr b=3`

This gave a very clean solution that `a=b=3`, making the factored quadratic:

`(x+3)(x+3)` or `color(red)((x+3)^2)`

Answer 2

See a solution process below:

Explanation:

Because the `x^2` coefficient is `1` we know the coefficient for the `x` terms in the factor will also be `1`:

`(x )(x )`

Because the constant is a positive and the coefficient for the `x` term is a positive we know the sign for the constants in the factors will both be positive because a positive plus a positive is a positive and positive times a positive is a positive:

`(x + )(x + )`

Now we need to determine the factors which multiply to 9 and also add to 6:

`1 xx 9 = 9`; `1 + 9 = 10` <- this is not the factor

`3 xx 3 = 9`; `3 + 3 = 6` <- this IS the factor

`(x + 3)(x + 3)`

Or

`(x + 3)^2`