Whakaoti i te 2x^2-70x+1225=0

$$2 x ^ { 2 } - 70 x + 1225 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{35}{2}+\frac{35}{2}i=17.5+17.5i$$

$$x=\frac{35}{2}-\frac{35}{2}i=17.5-17.5i$$

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.

$$2x^{2}-70x+1225=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -70 mō b, me 1225 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 2\times 1225}}{2\times 2}$$

Pūrua -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 2\times 1225}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-70\right)±\sqrt{4900-8\times 1225}}{2\times 2}$$

Whakareatia -8 ki te 1225.

$$x=\frac{-\left(-70\right)±\sqrt{4900-9800}}{2\times 2}$$

Tāpiri 4900 ki te -9800.

$$x=\frac{-\left(-70\right)±\sqrt{-4900}}{2\times 2}$$

Tuhia te pūtakerua o te -4900.

$$x=\frac{-\left(-70\right)±70i}{2\times 2}$$

Ko te tauaro o -70 ko 70.

$$x=\frac{70±70i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{70±70i}{4}$$

Nā, me whakaoti te whārite x=\frac{70±70i}{4} ina he tāpiri te ±. Tāpiri 70 ki te 70i.

$$x=\frac{70+70i}{4}$$

Whakawehe 70+70i ki te 4.

$$x=\frac{35}{2}+\frac{35}{2}i$$

Nā, me whakaoti te whārite x=\frac{70±70i}{4} ina he tango te ±. Tango 70i mai i 70.

$$x=\frac{70-70i}{4}$$

Whakawehe 70-70i ki te 4.

$$x=\frac{35}{2}-\frac{35}{2}i$$

Kua oti te whārite te whakatau.

$$x=\frac{35}{2}+\frac{35}{2}i x=\frac{35}{2}-\frac{35}{2}i$$

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.

$$2x^{2}-70x+1225=0$$

Me tango 1225 mai i ngā taha e rua o te whārite.

$$2x^{2}-70x+1225-1225=-1225$$

Mā te tango i te 1225 i a ia ake anō ka toe ko te 0.

$$2x^{2}-70x=-1225$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-70x}{2}=-\frac{1225}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{70}{2}\right)x=-\frac{1225}{2}$$

Whakawehe -70 ki te 2.

$$x^{2}-35x=-\frac{1225}{2}$$

Whakawehea te -35, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{35}{2}. Nā, tāpiria te pūrua o te -\frac{35}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=-\frac{1225}{2}+\left(-\frac{35}{2}\right)^{2}$$

Pūruatia -\frac{35}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-35x+\frac{1225}{4}=-\frac{1225}{2}+\frac{1225}{4}$$

Tāpiri -\frac{1225}{2} ki te \frac{1225}{4} 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.

$$x^{2}-35x+\frac{1225}{4}=-\frac{1225}{4}$$

Tauwehea x^{2}-35x+\frac{1225}{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}.

$$\left(x-\frac{35}{2}\right)^{2}=-\frac{1225}{4}$$

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

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{-\frac{1225}{4}}$$

Whakarūnātia.

$$x-\frac{35}{2}=\frac{35}{2}i x-\frac{35}{2}=-\frac{35}{2}i$$

Me tāpiri \frac{35}{2} ki ngā taha e rua o te whārite.

$$x=\frac{35}{2}+\frac{35}{2}i x=\frac{35}{2}-\frac{35}{2}i$$