Whakaoti i te n^2-4019n+2009^2=0

$$n ^ { 2 } - 4019 n + 2009 ^ { 2 } = 0$$

Whakaoti mō n

$$n = \frac{3 \sqrt{893} + 4019}{2} \approx 2054.324658392$$

$$n = \frac{4019 - 3 \sqrt{893}}{2} \approx 1964.675341608$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -4019 mō b, me 4036081 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -4019.

$$n=\frac{-\left(-4019\right)±\sqrt{16152361-4\times 4036081}}{2}$$

Whakareatia -4 ki te 4036081.

$$n=\frac{-\left(-4019\right)±\sqrt{16152361-16144324}}{2}$$

Tāpiri 16152361 ki te -16144324.

$$n=\frac{-\left(-4019\right)±\sqrt{8037}}{2}$$

Tuhia te pūtakerua o te 8037.

$$n=\frac{-\left(-4019\right)±3\sqrt{893}}{2}$$

Ko te tauaro o -4019 ko 4019.

$$n=\frac{4019±3\sqrt{893}}{2}$$

Nā, me whakaoti te whārite n=\frac{4019±3\sqrt{893}}{2} ina he tāpiri te ±. Tāpiri 4019 ki te 3\sqrt{893}.

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

Nā, me whakaoti te whārite n=\frac{4019±3\sqrt{893}}{2} ina he tango te ±. Tango 3\sqrt{893} mai i 4019.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

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

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

Tangohia te 4036081 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Whakawehea te -4019, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4019}{2}. Nā, tāpiria te pūrua o te -\frac{4019}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-4019n+\left(-\frac{4019}{2}\right)^{2}=-4036081+\left(-\frac{4019}{2}\right)^{2}$$

Pūruatia -\frac{4019}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-4019n+\frac{16152361}{4}=-4036081+\frac{16152361}{4}$$

Tāpiri -4036081 ki te \frac{16152361}{4}.

$$n^{2}-4019n+\frac{16152361}{4}=\frac{8037}{4}$$

Tauwehea n^{2}-4019n+\frac{16152361}{4}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{4019}{2}\right)^{2}=\frac{8037}{4}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(n-\frac{4019}{2}\right)^{2}}=\sqrt{\frac{8037}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{4019}{2} ki ngā taha e rua o te whārite.

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