I was given the problem $144x^4-121x^2y^2+16y^4.$ I used completing the square and got an answer of $12x^2+4y^2+11ixy$, I would like to know if this is correct.
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$\begingroup$ That is correct $\endgroup$– JohnColtraneisJCCommented Sep 11, 2017 at 12:35
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$\begingroup$ Have you tried checking your own work first by multiplying out your answer (which, presumably, is actually $(12x^2+4y^2+11ixy)^2$) to see if you get the original expression? $\endgroup$– amdCommented Sep 12, 2017 at 1:27
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$\begingroup$ It looks to me like you simply took to square roots of all of the coefficients. $\endgroup$– amdCommented Sep 13, 2017 at 1:54
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2 Answers
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try the Ansatz $$(12x^2-...-4y^2)(12x^2+...-4y^2)$$
answered Sep 11, 2017 at 12:37
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Hint
It is not correct.
Try this one:
Complete square for the terms $144x^4$ and $16y^4$ and get $$144x^4+ 16y^4=(12x^2-4y^2)^2+96x^2y^2$$
now use the whole thing
$$144x^4-121x^2y^2+16y^4=(12x^2-4y^2)^2+96x^2y^2-121x^2y^2=(12x^2-4y^2)^2-(5xy)^2$$
Can you finish?
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$\begingroup$ where did the *5xy)^2 come from? $\endgroup$ Commented Sep 11, 2017 at 12:40
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$\begingroup$ @user370026: take another look. It is clear? $\endgroup$– ArnaldoCommented Sep 11, 2017 at 12:43
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$\begingroup$ i would like to know why you added 96x^2y^2 $\endgroup$ Commented Sep 11, 2017 at 12:47
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$\begingroup$ @user370026: I am completing square. $\endgroup$– ArnaldoCommented Sep 11, 2017 at 12:49
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