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The-two-corner-points-of-a-square-lie-on-curve-f-x-x-2-2x-3-and-the-other-two-corner-points-lie-on-curve-g-x-x-2-2x-3-It-is-known-that-the-area-of-a-square-can-be-expressed-by-p-




Question Number 213522 by efronzo1 last updated on 07/Nov/24
  The two corner points of a square     lie on curve f(x)= x^2 −2x−3 and    the other two corner points lie on    curve g(x)= −x^2 +2x+3 . It is known   that the area of a square can be     expressed by p+q(√r) , for a natural    number p,q ,r where r is not divisible    by any perfect square number other 1.   The value of p+q+r =?
$$\:\:\mathrm{The}\:\mathrm{two}\:\mathrm{corner}\:\mathrm{points}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\: \\ $$$$\:\:\mathrm{lie}\:\mathrm{on}\:\mathrm{curve}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{3}\:\mathrm{and}\: \\ $$$$\:\mathrm{the}\:\mathrm{other}\:\mathrm{two}\:\mathrm{corner}\:\mathrm{points}\:\mathrm{lie}\:\mathrm{on}\: \\ $$$$\:\mathrm{curve}\:\mathrm{g}\left(\mathrm{x}\right)=\:−\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{3}\:.\:\mathrm{It}\:\mathrm{is}\:\mathrm{known} \\ $$$$\:\mathrm{that}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{can}\:\mathrm{be}\: \\ $$$$\:\:\mathrm{expressed}\:\mathrm{by}\:\mathrm{p}+\mathrm{q}\sqrt{\mathrm{r}}\:,\:\mathrm{for}\:\mathrm{a}\:\mathrm{natural}\: \\ $$$$\:\mathrm{number}\:\mathrm{p},\mathrm{q}\:,\mathrm{r}\:\mathrm{where}\:\mathrm{r}\:\mathrm{is}\:\mathrm{not}\:\mathrm{divisible}\: \\ $$$$\:\mathrm{by}\:\mathrm{any}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{number}\:\mathrm{other}\:\mathrm{1}. \\ $$$$\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{p}+\mathrm{q}+\mathrm{r}\:=? \\ $$

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