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Question Number 49186 by afachri last updated on 04/Dec/18

$$\mathbf{please}\mathbf{help}...$$$$$$$$\mathrm{There}\mathrm{is}:$$$$21{\mathit{x}}^2-21\mathit{p}\mathit{x}+49\mathit{p}-7=0$$$$\mathrm{whose}\mathrm{roots}\mathit{u}\mathrm{and}\mathit{v}.\mathrm{If}\mathit{u}\mathrm{and}\mathit{v}\mathrm{are}\mathrm{not}\in \mathbb{Z},$$$$\mathrm{and}\mathit{u},\mathit{v}⩾1.$$$$\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\mathit{u}+\mathit{v}!$$$$$$

Commented by MJS last updated on 04/Dec/18

$$\mathrm{strange}\mathrm{question}...{\mathrm{there}}^{\prime }\mathrm{s}\mathrm{not}\mathrm{a}\mathrm{single}\mathrm{solution}$$

Commented by afachri last updated on 04/Dec/18

$$\mathrm{i}\mathrm{thought}\mathrm{it}\mathrm{is}.\mathrm{i}\mathrm{found}\mathrm{this}\mathrm{question}\mathrm{in}\mathrm{test}\mathrm{of}\mathrm{University}.$$

Commented by mr W last updated on 04/Dec/18

$$x=\frac{21p\pm \sqrt{D}}{2\times 21}$$$$condition1:$$$$D={\left(21p\right)}^2-4\times 21\times (49p-7)⩾0$$$$3p^2-4(7p-1)⩾0$$$$3p^2-28p+4⩾0$$$$p⩽\frac{14-2\sqrt{46}}{3}=0.145$$$$p⩾\frac{14+2\sqrt{46}}{3}=9.188$$$$condition2:$$$$x_1=\frac{21p-\sqrt{D}}{2\times 21}⩾1$$$$21p-\sqrt{D}⩾42$$$$\sqrt{D}⩽21p-42$$$$D={\left(21p\right)}^2-4\times 21\times (49p-7)⩽{(21p-42)}^2$$$$1⩽p$$$$$$$$u+v=\frac{21p}{21}=p⩾2$$$$uv=\frac{49p-7}{21}=\frac{7p-1}{3}⩾1\Rightarrow p⩾\frac{4}{7}$$$$\Rightarrow u+v=p⩾\frac{14+2\sqrt{46}}{3}=9.188$$

Commented by afachri last updated on 04/Dec/18

$$\mathrm{nothing}\mathrm{matches}\mathrm{to}\mathrm{the}\mathrm{avaible}\mathrm{answers}.\mathrm{the}\mathrm{options}\mathrm{are}$$$$\mathrm{a}.\frac{59}{3}\mathrm{b}.\frac{60}{3}\mathrm{c}.\frac{61}{3}\mathrm{d}.\frac{62}{3}\mathrm{e}.\frac{63}{3}$$

Answered by MJS last updated on 04/Dec/18

$$x^2-px+\frac{7p-1}{3}=0$$$$x=\frac{p}{2}\pm \sqrt{\frac{3p^2-28p+4}{12}}\Rightarrow p⩽\frac{14-2\sqrt{46}}{3}\vee p⩾\frac{14+2\sqrt{46}}{3}$$$$u+v=p\mathrm{with}p⩾\frac{14+2\sqrt{46}}{3}\wedge p\ne 13$$

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