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Question Number 81610 by TawaTawa last updated on 14/Feb/20

Answered by MJS last updated on 14/Feb/20

$$2⩽x⩽4$$$$\mathrm{let}x=t^2+2\Rightarrow 0⩽t^2⩽2$$$$3t-12\sqrt{2-t^2}+21t^2-40+t\sqrt{2-t^2}=0$$$$21t^2+3t-40=(12-t)\sqrt{2-t^2}$$$${(21t^2+3t-40)}^2={(12-t^2)}^2(2-t^2)$$$$\Rightarrow$$$$t^4+\frac{3}{13}{t}^3-\frac{1529}{442}{t}^2-\frac{96}{221}t+\frac{656}{221}=0$$$$(t^2-\frac{32}{17})(t^2+\frac{3}{13}t-\frac{41}{26})=0$$$$t_{1,2}=\pm \frac{4\sqrt{34}}{17}$$$$t_{3,4}=-\frac{3}{26}\pm \frac{5\sqrt{43}}{26}$$$$0⩽t^2⩽2\Rightarrow \mathrm{all}{t}_j\mathrm{seem}\mathrm{to}\mathrm{fit}$$$$\mathrm{checking}\mathrm{all}x=t^2+2\mathrm{in}\mathrm{given}\mathrm{equation}$$$$\Rightarrow x=\frac{66}{17}$$

Commented by TawaTawa last updated on 14/Feb/20

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by MJS last updated on 14/Feb/20

$$3\sqrt{\frac{66}{17}-2}-12\sqrt{4-\frac{66}{17}}+21\times \frac{66}{17}-82+\sqrt{(4-\frac{66}{17})(\frac{66}{17}-2)}=$$$$=\frac{12\sqrt{34}}{17}-\frac{12\sqrt{34}}{17}-\frac{8}{17}+\frac{8}{17}=0$$

Commented by TawaTawa last updated on 14/Feb/20

$$\mathrm{Great}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{Seen}\mathrm{my}\mathrm{mistake}.$$

Commented by TawaTawa last updated on 14/Feb/20

$$\mathrm{Great}.\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{Seen}\mathrm{my}\mathrm{mistake}.$$

Commented by TawaTawa last updated on 14/Feb/20

$$\mathrm{Sir},\mathrm{the}\mathrm{answer}\mathrm{is}\mathrm{not}\mathrm{satisfy}.\mathrm{or}\mathrm{i}\mathrm{cannot}\mathrm{check}.$$

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