Whakaoti i te n^2-4019n+2009^2leq0

$$n ^ { 2 } - 4019 n + 2009 ^ { 2 } \leq 0$$

Whakaoti mō n

$$n\in \begin{bmatrix}\frac{4019-3\sqrt{893}}{2},\frac{3\sqrt{893}+4019}{2}\end{bmatrix}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 2009 mā te pū o 2, kia riro ko 4036081.

$$n^{2}-4019n+4036081\leq 0$$

Kia whakaotia te koreōrite, me tauwehe te taha mauī. Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$n^{2}-4019n+4036081=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 1 mō te a, te -4019 mō te b, me te 4036081 mō te c i te ture pūrua.

$$n=\frac{-\left(-4019\right)±\sqrt{\left(-4019\right)^{2}-4\times 1\times 4036081}}{2}$$

Mahia ngā tātaitai.

$$n=\frac{4019±3\sqrt{893}}{2}$$

Whakaotia te whārite n=\frac{4019±3\sqrt{893}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$n=\frac{3\sqrt{893}+4019}{2} n=\frac{4019-3\sqrt{893}}{2}$$

Tuhia anō te koreōrite mā te whakamahi i ngā otinga i whiwhi.

$$\left(n-\frac{3\sqrt{893}+4019}{2}\right)\left(n-\frac{4019-3\sqrt{893}}{2}\right)\leq 0$$

Kia ≤0 te otinga, me ≥0 rawa tētahi uara o n-\frac{3\sqrt{893}+4019}{2} me n-\frac{4019-3\sqrt{893}}{2}, me ≤0 anō te uara o tētahi. Whakaarohia te tauira ina ko n-\frac{3\sqrt{893}+4019}{2}\geq 0 me n-\frac{4019-3\sqrt{893}}{2}\leq 0.

$$n-\frac{3\sqrt{893}+4019}{2}\geq 0 n-\frac{4019-3\sqrt{893}}{2}\leq 0$$

He teka tēnei mō tētahi n ahakoa.

$$n\in \emptyset$$

Whakaarohia te tauira ina ko n-\frac{3\sqrt{893}+4019}{2}\leq 0 me n-\frac{4019-3\sqrt{893}}{2}\geq 0.

$$n-\frac{4019-3\sqrt{893}}{2}\geq 0 n-\frac{3\sqrt{893}+4019}{2}\leq 0$$

Te otinga e whakaea i ngā koreōrite e rua ko n\in \left[\frac{4019-3\sqrt{893}}{2},\frac{3\sqrt{893}+4019}{2}\right].

$$n\in \begin{bmatrix}\frac{4019-3\sqrt{893}}{2},\frac{3\sqrt{893}+4019}{2}\end{bmatrix}$$

Ko te otinga whakamutunga ko te whakakotahi i ngā otinga kua whiwhi.

$$n\in \begin{bmatrix}\frac{4019-3\sqrt{893}}{2},\frac{3\sqrt{893}+4019}{2}\end{bmatrix}$$