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There is an expression: $(h+2p)^2(x+2y)^2 - 3p^2(x+2y)^2-3y^2(h+2p)^2+9p^2y^2$

Is there a way to simplify this to the form $l^2 + 4lk + k^2$? I tried to substitute $l = (h+2p)(x+2y)$ and $k= 3py$ and this takes care of the expressions at the two ends, but the expressions in the middle are stumping me.

Alternatively, one may try expressing it in the form $(l+2k)^2 - 3lk$, but I am unable to do this as well. However, I suspect that there is some relationship as $-3lk$ bears resemblance to $- 3p^2(x+2y)^2-3y^2(h+2p)^2$ if you factor out the $-3$. Is it possible to use Fibonnacci Identity in this endeavor?

EDIT: The original expression is $(h^2 + 4ph + p^2)(x^2 + 4xy + y^2)$ I converted this to $$((h+2p)^2-3p^2)((x+2y)^2-3y^2)$$

EDIT: I tried an alternative approach. I multiplied out this expression: $$(h^2 + 4ph + p^2)(x^2 + 4xy + y^2)$$

and got the following: $$h^2x^2 + p^2x^2 + h^2y^2 + p^2y^2 + 4hpx^2 + 4xyh^2 + 4xyp^2 +4hpy^2 + 16hpxy$$

(I'm leaving out some steps) This can be factored into : $$(hx + py)^2+4(px+hy)(hx+py)+(px+hy)^2 +12hpxy$$

This would have been perfect except for the pesky $12hpxy$.

Τίμων
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3 Answers3

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Answer: $$(h x-p y)^2 + 4(h x-p y)(h y+p x+4 p y) +(h y+p x+4 p y)^2 = (h^2 + 4p h + p^2)(x^2 + 4x y + y^2) $$

How I found this result:
Rewrite $$(h^2 + 4p h + p^2)(x^2 + 4x y + y^2) = ((h+2p)^2-3p^2)((x+2y)^2-3y^2) = (A^2-3B^2)(C^2-3D^2)$$ where $A=h+2p$, $B=p$, $C=x+2y$ and $D=y$. Now, we have $$(A^2-3B^2)(C^2-3D^2) = (AC+3BD)^2 - 3(AD + BC)^2 = \lbrace[AC+3BD-2(AD+BC)]+2(AD+BC)\rbrace^2 - 3(AD + BC)^2 $$ Let $l=AC+3BD-2AD-2BC$ and $k=AD + BC$. Now, $(l+2k)^2-3k^2=l^2 +4lk + k^2 $, which is what we wanted. Finally, substitute $A$, $B$, $C$ and $D$.

Τίμων
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  • Why the downvote? Is it wrong? Have I misunderstood the question? – Τίμων Mar 09 '16 at 15:46
  • Could you elaborate, show everything you did? –  Mar 10 '16 at 04:38
  • Yes, I think you may have misunderstood the question. You see, $(AC + 3BD)^2$ does not help us. If it were $$(AC + 2BD)^2$$ then it would be helpful. Furthermore you cannot use the conversion of $$(l + 2k)^2 - 3k^2$$ to $l^2 + 4lk + k^2$ because $(AD + BC)$ is not the same as $3BD$. –  Mar 10 '16 at 04:46
  • I've edited the answer so as to make it clearer. You should see a reply to your comment there. :) – Τίμων Mar 10 '16 at 07:02
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Combining the first two terms gives $(x+2y)^2(h^2+4ph+p^2)$, and combining the second two terms gives $-3y^2(h^2+4ph+p^2)$. Combining these two together gives $$(x^2+4xy+y^2)(h^2+4ph+p^2).$$ I don't see a way to get the form you described, but at least this is a product of two things of that form.

Michael Harrison
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  • Thanks for the answer, but the problem depends on the terms being expressed in that form. –  Mar 02 '16 at 10:18
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Question: Simplify the expression $(h^2 + 4hp + p^2)(x^2 + 4xy + y^2)$ in the form of $l^2 + 4lk + k^2$.

Working Out

Note that $l^2 + 4lk + k^2 = (l + 2k)^2 - 3k^2$.

Simplify expression to: $$((h + 2p)^2 - 3p^2)((x + 2y)^2 - 3y^2)$$ Then expand using FOIL: $$(h+2p)^2(x+2y)^2 - 3y^2(h+2p)^2 - 3p^2(x+2y)^2 + 9p^2y^2$$

Then define $a=h+2p$ and $b=x+2y$, and simplify: $$(ab)^2 - 3(y^2a^2 + p^2b^2 - 3p^2y^2)$$

This form is very similar to $(l + 2k)^2 - 3k^2$

So, $$(ab)^2 = (l+2k)^2$$ $$k^2 = y^2a^2 + p^2b^2 - 3p^2y^2$$

Simplify to: $$k = \sqrt {y^2a^2 + p^2b^2 - 3p^2y^2}$$ $$l = ab - 2(\sqrt {y^2a^2 + p^2b^2 - 3p^2y^2})$$

Sub $a=h+2p$ and $b=x+2y$,

$$\therefore k = \sqrt {y^2(h+2p)^2 + p^2(x+2y)^2 - 3p^2y^2}$$ $$\therefore l = (h+2p)(x+2y) - 2(\sqrt {y^2(h+2p)^2 + p^2(x+2y)^2 - 3p^2y^2})$$

frog1944
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