SOLUTION: Completing the square 1. X^4+64y^4 2. X^4+y^2+25 3. X^4-11x^2y^2+y^4 Please do help me, My teacher's solution goes like this, and I didn't even understand cuz he's too f

Question 892918: Completing the square
1. X^4+64y^4
2. X^4+y^2+25
3. X^4-11x^2y^2+y^4

Please do help me,
My teacher's solution goes like this, and I didn't even understand cuz he's too fast.
Ex. x^4+2x^2y^2+9y^4
x^4+9y^4+2x^2y^2
x^4+2x^2(3y^2)+9y^4+2x^2y^2-@x^2(3y^2)
X^4+6x^2y^2+9y^4+2x^2y^2-6x^2y^2
(x^2+3y^2)^2-4x^2y^2
(x^2+3y^2)^2-(2xy)^2
(x^2+3y^2+2xy)(x^2+3y^2-2xy)

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Look for terms that are squares. It is not easy to see them at first. It takes practice.

I will over-explain your teacher's solution:
In your teacher sees two squares: and .
The expressions and , squared, appear in .
If we add those two expressions and square the sum we have

To make the square appear,
your teacher does some plastic surgery to .
He takes something out of one end and adds it to another place, just like a plastic surgeon would.
He adds to "complete the square",
and subtracts the same so as not to really change anything.
Doing that to , he gets

Of course, he knows that
and thinks that is a more elegant way to write it so he writes

The first three terms are his completed square,

and now he writes that as for short,
and he should have written

Collecting like terms, he gets

Now he realizes that is also a square, and re-writes his expression as

He likes that because now he has a difference of squares,
and he knows that for any two expressions and .
So, with and

Of course, we do not need those parentheses around and around ,
I just wrote those parentheses so you would see the separate expressions.
So your teacher writes, without unnecessary parentheses,

1. (I found two squares that are added).
That could be part of the square of a sum:

To make the complete square
appear in , I add and subtract
to get .
I can re-write that as .
Then, since is a also a square, I have the difference of squares
, and I can re-write it as

For 2. and 3. , I see some squares, but I do not immediately see what to do with them.

2. has three squares: , , and

3. has the squares and .
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