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Question Number 20118 by Tinkutara last updated on 22/Aug/17

$$\mathrm{The}\mathrm{quadratic}\mathrm{equations}{x}^2-6x+a=0$$$$\mathrm{and}{x}^2-cx+6=0\mathrm{have}\mathrm{one}\mathrm{root}\mathrm{in}$$$$\mathrm{common}.\mathrm{The}\mathrm{other}\mathrm{roots}\mathrm{of}\mathrm{the}\mathrm{first}$$$$\mathrm{and}\mathrm{second}\mathrm{equations}\mathrm{are}\mathrm{integers}\mathrm{in}$$$$\mathrm{the}\mathrm{ratio}4:3.\mathrm{Then},\mathrm{find}\mathrm{the}\mathrm{common}$$$$\mathrm{root}.$$

Answered by ajfour last updated on 22/Aug/17

$$Letthecommonrootbe\alpha$$$$\alpha \beta =a,\alpha +\beta =6,\alpha \gamma =6,\alpha +\gamma =c$$$$\Rightarrow \beta -\gamma =6-c$$$$\frac{\beta }{\gamma }=\frac{a}{6}=\frac{4}{3}\Rightarrow a=8$$$$\alpha +\beta =6;\alpha \beta =8,\Rightarrow \alpha =2,4$$$$Acheck:$$$$if\alpha =2,then\beta =4,and\gamma =\frac{3}{4}\beta =3$$$$if\alpha =4,then\beta =2,and\gamma =\frac{3}{4}\beta =\frac{3}{2};$$$$notaninteger.$$$$Hence\alpha =2.c=\alpha +\gamma =2+3=5.$$

Commented by Tinkutara last updated on 22/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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