Question

Can anybody help me with Factoring Special Cases? I go to CCA!

Answer 1

@myininaya can you help??

Answer 2

What is the question

Answer 3

What is the factored form of q^2-12q+36?

Answer 4

This is a special case of "perfect square".
Do you know how to spot it?

Answer 5

No, I don't. This is honestly one of the most confusing things I've learned

Answer 6

It's a nice one to know, because it makes you life easier.
What you do is to following the following steps:
If the expression is a perfect square, it satisfies ALL of the following:
1. The quadratic term is a perfect square, e.g. q^2 = (q)^2, 4x^2 = (2x)^2, 81y^2=(9y)^2, etc.
so does you expression satisfy this condition?

Answer 7

No??

Answer 8

q^2 = (q)^2
so the square-root of q^2 is q, yes, the expression satisfies the first condition.
Still following?

Answer 9

Oh okay makes sense

Answer 10

Is that all??

Answer 11

Nope, step 2 is to check the same for the constant term (the term that does not have a variable).
For example, in
4x^2-20x+25
25 is the constant term.
We need to check if 25 is a perfect square. Can you do that for me?

Answer 12

Is it a perfect square?

Answer 13

If you still remember your times table, you will know that 5\(\times\)5=25
Since the two fives are identical, so 5^2=25, or sqrt(25)=5, so 25 is a perfect square.

If you have forgotten about the times table, it's time to review, and re-memorize it.
For help: https://www.mathsisfun.com/tables.html
Memorize at least until 10x10.

Now we come to step 3 to recognizing a perfect square polynomial.
So far, we have recognized the requiremenst for the first and last terms. Now the middle term.

If we have the first term as 49x^2, then its square-root is 7x, since (7x)^2=49x^2.
Similarly, if the last term is 81, then the square-root is 9, since 9^2=81.
Then
3. The middle term must be twice the product of the square-roots of the first and third terms, in this case, it must be 2*7x*9=18, or
we can see that
49x^2+18x+81 is a perfect square, and is equal to (7x+9)^2.

If the middle term is negative, then all we need to do is to put a minus sign in our answer, i.e.
49x^2-18x+81 is a perfect square, and is equal to (7x-9)^2.

To resume the steps:
1. Is the first term a perfect square?
2. Is the third term a perfect square?
3. Is the middle term equal to twice the product of the square-roots of the first and third terms?
If it is a yes to ALL three questions, we have an expression factorable using perfect squares.

Let do another example:

Factor, if possible, 25p^2+90pq+81q^2.
1. 25p^2=(5p)^2, so it is a perfect square.
2. 81q^2=(9q)^2, so it is a perfect square.
3. middle term 90pq=2*5p*9q=90pq, so satisfies the last step as well.
Therefore
25p^2+90pq+81q^2. is a perfect square, equal to (5p+9q)^2.