Question

I'll give medal!!! I need help on understanding how to factor quadratics. Here's the equation:
(x^2-4x+4)+(y^2-2y+1)=9. I know the answer but I don't understand how to get there. Plz help:)

Answer 1

When you have a trinomial that you need to factor, you need to check if it is a perfect square trinomial.

Answer 2

How do I do that?

Answer 3

Let's look at x^2 - 4x + 4
This may be (x - 2)^2, but we need to be sure.

Answer 4

1) The first term must be a perfect square.
If so, what is it the square of?
1) Answer: x^2 is the square of x.

Answer 5

Ok.

Answer 6

2) The last term must be a perfect square.
If so, what is it the square of?
2) Answer: 4 is a perfect square, and it is the square of 2 and -2.

Answer 7

That means this trinomial can possibly be (x + 2)^2 or (x - 2)^2.
Now we need to check the middle term of the trinomial to see if it works.

Answer 8

The middle term of a perfect square trinomial is in red below:

\( (a + b)^2 = a^2 \color{red}{+ 2ab} + b^2\)

\( (a - b)^2 = a^2 \color{red}{- 2ab} + b^2\)

Answer 9

So for it to be perfect, it has to be 2 times the first and last term? Positive or negative.

Answer 10

Exactly.

Answer 11

Once you know what the first and last terms of the TRINOMIAL are squares of, the middle term of the trinomial is 2 times the first term times the second term of the BINOMIAL.

Answer 12

So it would look like 2*(x*4)?

Answer 13

Back to our trinomial.
Since the middle term of the trinomial is negative, the only possibility is \((x - 2)^2\)

Answer 14

Back to our trinomial.
Since the middle term of the trinomial is negative, the only possibility is
(x - 2)^2

Answer 15

Got it

Answer 16

So then the answer would be the square (or what ever the power is) of what we now know is the middle term?