Question

How do you factor completely `x^2 - 8x + 16`?

Answer 1

This is a perfect square trinomial, because the first and last terms are perfect squares `(sqrt(x^2) = x and sqrt(16) = 4)`

Explanation:

Method 1:

`(x - 4)(x - 4)`

`(x - 4)^2`

You can also double check by making sure term b (the middle term) satisfies the equation `b = 2ac` only once you have factored, or when you have taken the square root of the first and last term. We check: `8 = 2(x)(4)`. So, we have factored the trinomial properly. Also, you can check by doing FOIL (first, outside, inside and last), multiplying out.

Method 2:

Factor as a regular trinomial of the form `ax^2 + bx + c, a = 1`. This method, although longer, is good to get used to because you will have to learn at one point to factor trinomials such as `x^2 + 8x + 15`, and it is the most safe and foolproof method.

To factor a trinomial of the form ax^2 + bx + c, a = 1#, you must find two numbers that multiply to c and that add to b.

We must find two numbers that multiply to +16 and add to -8. These two numbers are -4 and -4.

So, (x - 4)(x - 4). Since the parentheses repeats itself twice, we can rewrite the expression as `(x - 4)^2`

Practice exercises:

1. Factor the following trinomials using method 1

a) `x^2 + 10x + 25`

b) `16x^2 - 56x + 49`

2 . Factor the following trinomials using method 2

a) `x^2 - 22x + 121`

b) `x^2 + 5x + 6`

c) `x^2 - 8x - 33`

d) `x^2 - 14x + 45`

3 . Find the value of `m` that makes the following trinomials perfect square trinomials

a) `4x^2 + mx + 64`

b) `25x^2 - 40x + m`

Good luck!