Whakaoti i te 25x^2-90x+87=0

$$25 { x }^{ 2 } -90x+87=0$$

Whakaoti mō x (complex solution)

$$x=\frac{9+\sqrt{6}i}{5}\approx 1.8+0.489897949i$$

$$x=\frac{-\sqrt{6}i+9}{5}\approx 1.8-0.489897949i$$

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.

$$25x^{2}-90x+87=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, -90 mō b, me 87 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 25\times 87}}{2\times 25}$$

Pūrua -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 25\times 87}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-\left(-90\right)±\sqrt{8100-100\times 87}}{2\times 25}$$

Whakareatia -100 ki te 87.

$$x=\frac{-\left(-90\right)±\sqrt{8100-8700}}{2\times 25}$$

Tāpiri 8100 ki te -8700.

$$x=\frac{-\left(-90\right)±\sqrt{-600}}{2\times 25}$$

Tuhia te pūtakerua o te -600.

$$x=\frac{-\left(-90\right)±10\sqrt{6}i}{2\times 25}$$

Ko te tauaro o -90 ko 90.

$$x=\frac{90±10\sqrt{6}i}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{90±10\sqrt{6}i}{50}$$

Nā, me whakaoti te whārite x=\frac{90±10\sqrt{6}i}{50} ina he tāpiri te ±. Tāpiri 90 ki te 10i\sqrt{6}.

$$x=\frac{90+10\sqrt{6}i}{50}$$

Whakawehe 90+10i\sqrt{6} ki te 50.

$$x=\frac{9+\sqrt{6}i}{5}$$

Nā, me whakaoti te whārite x=\frac{90±10\sqrt{6}i}{50} ina he tango te ±. Tango 10i\sqrt{6} mai i 90.

$$x=\frac{-10\sqrt{6}i+90}{50}$$

Whakawehe 90-10i\sqrt{6} ki te 50.

$$x=\frac{-\sqrt{6}i+9}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{9+\sqrt{6}i}{5} x=\frac{-\sqrt{6}i+9}{5}$$

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.

$$25x^{2}-90x+87=0$$

Me tango 87 mai i ngā taha e rua o te whārite.

$$25x^{2}-90x+87-87=-87$$

Mā te tango i te 87 i a ia ake anō ka toe ko te 0.

$$25x^{2}-90x=-87$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}-90x}{25}=-\frac{87}{25}$$

Mā te whakawehe ki te 25 ka wetekia te whakareanga ki te 25.

$$x^{2}+\left(-\frac{90}{25}\right)x=-\frac{87}{25}$$

Whakahekea te hautanga \frac{-90}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{18}{5}x=-\frac{87}{25}$$

Whakawehea te -\frac{18}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{5}. Nā, tāpiria te pūrua o te -\frac{9}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{18}{5}x+\left(-\frac{9}{5}\right)^{2}=-\frac{87}{25}+\left(-\frac{9}{5}\right)^{2}$$

Pūruatia -\frac{9}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=\frac{-87+81}{25}$$

Tāpiri -\frac{87}{25} ki te \frac{81}{25} 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.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=-\frac{6}{25}$$

Tauwehea x^{2}-\frac{18}{5}x+\frac{81}{25}. 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}.

$$\left(x-\frac{9}{5}\right)^{2}=-\frac{6}{25}$$

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

$$\sqrt{\left(x-\frac{9}{5}\right)^{2}}=\sqrt{-\frac{6}{25}}$$

Whakarūnātia.

$$x-\frac{9}{5}=\frac{\sqrt{6}i}{5} x-\frac{9}{5}=-\frac{\sqrt{6}i}{5}$$

Me tāpiri \frac{9}{5} ki ngā taha e rua o te whārite.

$$x=\frac{9+\sqrt{6}i}{5} x=\frac{-\sqrt{6}i+9}{5}$$