Whakaoti i te 9D^2-245D+500=0

$$9 D ^ { 2 } - 245 D + 500 = 0$$

Whakaoti mō D

$$D = \frac{20}{9} = 2\frac{2}{9} \approx 2.222222222$$

$$D=25$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$9D^{2}-245D+500=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -245 mō b, me 500 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$D=\frac{-\left(-245\right)±\sqrt{\left(-245\right)^{2}-4\times 9\times 500}}{2\times 9}$$

Pūrua -245.

$$D=\frac{-\left(-245\right)±\sqrt{60025-4\times 9\times 500}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$D=\frac{-\left(-245\right)±\sqrt{60025-36\times 500}}{2\times 9}$$

Whakareatia -36 ki te 500.

$$D=\frac{-\left(-245\right)±\sqrt{60025-18000}}{2\times 9}$$

Tāpiri 60025 ki te -18000.

$$D=\frac{-\left(-245\right)±\sqrt{42025}}{2\times 9}$$

Tuhia te pūtakerua o te 42025.

$$D=\frac{-\left(-245\right)±205}{2\times 9}$$

Ko te tauaro o -245 ko 245.

$$D=\frac{245±205}{2\times 9}$$

Whakareatia 2 ki te 9.

$$D=\frac{245±205}{18}$$

Nā, me whakaoti te whārite D=\frac{245±205}{18} ina he tāpiri te ±. Tāpiri 245 ki te 205.

$$D=\frac{450}{18}$$

Whakawehe 450 ki te 18.

$$D=25$$

Nā, me whakaoti te whārite D=\frac{245±205}{18} ina he tango te ±. Tango 205 mai i 245.

$$D=\frac{40}{18}$$

Whakahekea te hautanga \frac{40}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$D=\frac{20}{9}$$

Kua oti te whārite te whakatau.

$$D=25 D=\frac{20}{9}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$9D^{2}-245D+500=0$$

Me tango 500 mai i ngā taha e rua o te whārite.

$$9D^{2}-245D+500-500=-500$$

Mā te tango i te 500 i a ia ake anō ka toe ko te 0.

$$9D^{2}-245D=-500$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9D^{2}-245D}{9}=-\frac{500}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$D^{2}-\frac{245}{9}D=-\frac{500}{9}$$

Whakawehea te -\frac{245}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{245}{18}. Nā, tāpiria te pūrua o te -\frac{245}{18} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$D^{2}-\frac{245}{9}D+\left(-\frac{245}{18}\right)^{2}=-\frac{500}{9}+\left(-\frac{245}{18}\right)^{2}$$

Pūruatia -\frac{245}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$D^{2}-\frac{245}{9}D+\frac{60025}{324}=-\frac{500}{9}+\frac{60025}{324}$$

Tāpiri -\frac{500}{9} ki te \frac{60025}{324} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$D^{2}-\frac{245}{9}D+\frac{60025}{324}=\frac{42025}{324}$$

Tauwehea D^{2}-\frac{245}{9}D+\frac{60025}{324}. 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}.

$$\left(D-\frac{245}{18}\right)^{2}=\frac{42025}{324}$$

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

$$\sqrt{\left(D-\frac{245}{18}\right)^{2}}=\sqrt{\frac{42025}{324}}$$

Whakarūnātia.

$$D-\frac{245}{18}=\frac{205}{18} D-\frac{245}{18}=-\frac{205}{18}$$

Me tāpiri \frac{245}{18} ki ngā taha e rua o te whārite.

$$D=25 D=\frac{20}{9}$$