Solve the equation $$\sqrt{x^2-3x+18}+\sqrt{x^2+5x+2} = 2\sqrt{x^4+2x^3+5x^2+84x+36}.$$ I don't know where to start, but what I know is that $$(x^2-3x+18)(x^2+5x+2) = x^4+2x^3+5x^2+84x+36.$$ Any suggestion would be appreciate.
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@AnotherUser don’t you think the context is enough? The OP has made a key observation wrt factoring the quartic polynomial – insipidintegrator Jul 05 '22 at 13:42
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I'll give you a hint. I think you already know about this, but $(x^2-3x+18)(x^2+5x+2)=x^4+2x^3+5x^2+84x+36$. So, let $\sqrt{x^2-3x+18}=A, \sqrt{x^2+5x+2}=B$. So, $A+B=2AB.$ You may start from here. – RDK Jul 05 '22 at 13:43
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@wasn'tme it's not $(a+b)/2 = \sqrt{ab}$, it's $\sqrt{a}+\sqrt{b} = 2\sqrt{ab}$ – Longirsu Jul 05 '22 at 13:44
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@wasn'tme It is, $(a+b)/2=ab$. – RDK Jul 05 '22 at 13:44
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@RDK ok, lemme try a sec – Longirsu Jul 05 '22 at 13:44
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@RDK still don't figure it out, can you give me more hint T0T – Longirsu Jul 05 '22 at 13:48
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Alright, second hint. $2A+2B=4AB, 4AB-2A-2B=0.$ Try adding some numbers to both sides, and make this to $(\square)(\square)=\square$. – RDK Jul 05 '22 at 13:50
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Ok, let me try. – Longirsu Jul 05 '22 at 13:51
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Let us continue this discussion in chat. – RDK Jul 05 '22 at 13:55
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Still didn't figure it out, any help would be appreciate! – Longirsu Jul 05 '22 at 14:25
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Problem leads to 8th degree equation with integer coefficients. I suppose there is no closed-form solution. – Ivan Kaznacheyeu Jul 05 '22 at 16:25
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@IvanKaznacheyeu TvT – Longirsu Jul 06 '22 at 01:25