explanation of completing the squares

Question

Is it possible for someone explain to me the process of completing the squares? I've been having confusions about it.

please explain using this problem: 1) $144x^4-121x^2y^2+16y^4$

thank you!

Answer 1

The first thing you might do is try expanding $(12x^2 - 4y^2)^2$. When you do you get $144 x^4 - 96 x^2 y^2 + 16 y^4$. Darn it anyway!

The process of completing the square is usually all about deriving the quadratic formula, but it can be directly used to solve 'quadratic type stuff'. So if you can't simply plug into the quadratic formula, port the proof of the quadratic formula to the problem $144x^4-121x^2y^2+16y^4$ posed here. However, there are two techniques for completing the square - see this.

Clarification: The quadratic formula tells us how to factor a quadratic equation in $x$ into two linear factors. The OP's question is (after some thought) about factoring a $4^{th}$ 'quadratic-like' degree equation in $x$ and $y$ into two $2^{nd}$ 'quadratic-like' degree equations in those variables. In some cases you might be able to factor further, for example, $x^2 + 2xy + y^2 = (x+y)^2$.

It won't always be so pretty of a picture, but Robert Z directly demonstrates 'how to go at it' by completing the square.

Answer 2

Hint. Note that $$X^2+Y^2=X^2+Y^2-2XY+2XY=(X-Y)^2+2XY$$ (by subtracting $2XY$ we "complete" $X^2+Y^2$ to the square $(X-Y)^2$).

In your case let $X=12x^2$ and $Y=4y^2$, then $$144x^4-121x^2y^2+16y^4=(12x^2-4y^2)^2+96x^2y^2-121x^2y^2= (12x^2-4y^2)^2-(5xy)^2$$ Are you able to complete the factorization?