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y-x-2-4x-9-x-2-4x-5-the-values-of-the-feture-set-of-prime-numbers-




Question Number 12632 by @ANTARES_VY last updated on 27/Apr/17
y=((x^2 −4x+9)/(x^2 −4x+5))   the  values  of the  feture  set  of  prime  numbers?
$$\boldsymbol{\mathrm{y}}=\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{x}}+\mathrm{9}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{\mathrm{x}}+\mathrm{5}}\:\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{values}}\:\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{feture}}\:\:\boldsymbol{\mathrm{set}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{prime}}\:\:\boldsymbol{\mathrm{numbers}}? \\ $$
Answered by abcd last updated on 27/Apr/17
Commented by prakash jain last updated on 28/Apr/17
There is a mistake in calculation  of determinant above.  Discriminant  D=b^2 −4ac  =4^2 (y−1)^2 −4(y−1)(5y−9)  =4(y−1)(−y+5)  =−4(y−1)(y−5)  D≥0⇒1≤y≤5  y=1 is not possible original  function except in limiting case  of x→∞  1<y≤5
$$\mathrm{There}\:\mathrm{is}\:\mathrm{a}\:\mathrm{mistake}\:\mathrm{in}\:\mathrm{calculation} \\ $$$$\mathrm{of}\:\mathrm{determinant}\:\mathrm{above}. \\ $$$$\mathrm{Discriminant} \\ $$$$\mathrm{D}={b}^{\mathrm{2}} −\mathrm{4}{ac} \\ $$$$=\mathrm{4}^{\mathrm{2}} \left({y}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{4}\left({y}−\mathrm{1}\right)\left(\mathrm{5}{y}−\mathrm{9}\right) \\ $$$$=\mathrm{4}\left({y}−\mathrm{1}\right)\left(−{y}+\mathrm{5}\right) \\ $$$$=−\mathrm{4}\left({y}−\mathrm{1}\right)\left({y}−\mathrm{5}\right) \\ $$$${D}\geqslant\mathrm{0}\Rightarrow\mathrm{1}\leqslant{y}\leqslant\mathrm{5} \\ $$$${y}=\mathrm{1}\:\mathrm{is}\:\mathrm{not}\:\mathrm{possible}\:\mathrm{original} \\ $$$$\mathrm{function}\:\mathrm{except}\:\mathrm{in}\:\mathrm{limiting}\:\mathrm{case} \\ $$$$\mathrm{of}\:{x}\rightarrow\infty \\ $$$$\mathrm{1}<{y}\leqslant\mathrm{5} \\ $$

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