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Question Number 70996 by A8;15: last updated on 10/Oct/19

Commented by mathmax by abdo last updated on 10/Oct/19

$$lettakeatry\left(e\right)\Rightarrow \sqrt{x^3+24}-\sqrt{x^3+12}=3x+8\Rightarrow$$$${(\sqrt{x^3+24}-\sqrt{x^3+12})}^2=9x^2+48x+64\Rightarrow$$$$x^3+24-2\sqrt{(x^3+24)(x^3+12)}+x^3+12=9x^2+48x+64\Rightarrow$$$$2x^3+36-9x^2-48x-64=2\sqrt{(....)(...)}\Rightarrow$$$$2x^3-9x^2-28=2\sqrt{(x^3+24)(x^3+12})\Rightarrow$$$${(2x^3-9x^2-28)}^2=4(x^6+12x^3+24x^3+24\times 12)\Rightarrow$$$${(2x^3-9x^2)}^2-56(2x^3-9x^2)+{28}^2=4x^6+174x^3+4\times 12\times 24..$$$$thatleadtoequationof{5}^{eme}degre....$$

Commented by MJS last updated on 10/Oct/19

$$\mathrm{yes}\mathrm{indeed}.\mathrm{and}\mathrm{trying}\mathrm{the}\mathrm{factors}\mathrm{of}\mathrm{the}$$$$\mathrm{constant}\mathrm{we}\mathrm{find}x=-2$$$$\mathrm{the}\mathrm{other}\mathrm{four}\mathrm{solutions}\mathrm{are}\mathrm{wrong}(\mathrm{we}\mathrm{had}$$$$\mathrm{to}\mathrm{square}2\mathrm{times})$$

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