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Question Number 117082 by bobhans last updated on 09/Oct/20

$$\mathrm{If}6-3\mathrm{x}\mathrm{is}\mathrm{the}\mathrm{geometric}\mathrm{mean}\mathrm{between}$$$$\mathrm{the}\mathrm{integer}{\mathrm{x}}^2+2\mathrm{and}\overline{2}\mathrm{what}\mathrm{are}\mathrm{the}$$$$\mathrm{values}\mathrm{of}\mathrm{x}?$$

Answered by bemath last updated on 09/Oct/20

$$\mathrm{geometric}\mathrm{mean}\Rightarrow \sqrt{({\mathrm{x}}^2+2).2}=6-3\mathrm{x}$$$$\mathrm{the}\mathrm{equation}\mathrm{valid}\mathrm{for}6-3\mathrm{x}⩾0$$$$\mathrm{or}\mathrm{x}⩽2.$$$$\mathrm{squaring}\mathrm{both}\mathrm{sides}\mathrm{gives}$$$${2\mathrm{x}}^2+4={9\mathrm{x}}^2-36\mathrm{x}+36$$$${7\mathrm{x}}^2-36\mathrm{x}+32=0$$$$(7\mathrm{x}-8)(\mathrm{x}-4)=0$$$$\{\begin{array}{l}\mathrm{x}=\frac{8}{7}\leftarrow \mathrm{accept}\\ \mathrm{x}=4\leftarrow \mathrm{rejected}\end{array}$$$$\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{x}\mathrm{is}\frac{8}{7}.$$

Commented by bemath last updated on 09/Oct/20

$$\mathrm{i}\mathrm{think}\mathrm{this}\mathrm{question}\mathrm{not}\mathrm{consisten}$$$$\mathrm{because}\mathrm{x}=\frac{8}{7}\mathrm{not}\mathrm{a}\mathrm{integer}$$

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