Whakaoti i te 5x^2-16x+29=0

$$5 { x }^{ 2 } -16x+29=0$$

Whakaoti mō x (complex solution)

$$x=\frac{8}{5}+\frac{9}{5}i=1.6+1.8i$$

$$x=\frac{8}{5}-\frac{9}{5}i=1.6-1.8i$$

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.

$$5x^{2}-16x+29=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -16 mō b, me 29 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 5\times 29}}{2\times 5}$$

Pūrua -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 5\times 29}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-16\right)±\sqrt{256-20\times 29}}{2\times 5}$$

Whakareatia -20 ki te 29.

$$x=\frac{-\left(-16\right)±\sqrt{256-580}}{2\times 5}$$

Tāpiri 256 ki te -580.

$$x=\frac{-\left(-16\right)±\sqrt{-324}}{2\times 5}$$

Tuhia te pūtakerua o te -324.

$$x=\frac{-\left(-16\right)±18i}{2\times 5}$$

Ko te tauaro o -16 ko 16.

$$x=\frac{16±18i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{16±18i}{10}$$

Nā, me whakaoti te whārite x=\frac{16±18i}{10} ina he tāpiri te ±. Tāpiri 16 ki te 18i.

$$x=\frac{16+18i}{10}$$

Whakawehe 16+18i ki te 10.

$$x=\frac{8}{5}+\frac{9}{5}i$$

Nā, me whakaoti te whārite x=\frac{16±18i}{10} ina he tango te ±. Tango 18i mai i 16.

$$x=\frac{16-18i}{10}$$

Whakawehe 16-18i ki te 10.

$$x=\frac{8}{5}-\frac{9}{5}i$$

Kua oti te whārite te whakatau.

$$x=\frac{8}{5}+\frac{9}{5}i x=\frac{8}{5}-\frac{9}{5}i$$

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.

$$5x^{2}-16x+29=0$$

Me tango 29 mai i ngā taha e rua o te whārite.

$$5x^{2}-16x+29-29=-29$$

Mā te tango i te 29 i a ia ake anō ka toe ko te 0.

$$5x^{2}-16x=-29$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-16x}{5}=-\frac{29}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{16}{5}x=-\frac{29}{5}$$

Whakawehea te -\frac{16}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{8}{5}. Nā, tāpiria te pūrua o te -\frac{8}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{16}{5}x+\left(-\frac{8}{5}\right)^{2}=-\frac{29}{5}+\left(-\frac{8}{5}\right)^{2}$$

Pūruatia -\frac{8}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{16}{5}x+\frac{64}{25}=-\frac{29}{5}+\frac{64}{25}$$

Tāpiri -\frac{29}{5} ki te \frac{64}{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.

$$x^{2}-\frac{16}{5}x+\frac{64}{25}=-\frac{81}{25}$$

Tauwehea x^{2}-\frac{16}{5}x+\frac{64}{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}.

$$\left(x-\frac{8}{5}\right)^{2}=-\frac{81}{25}$$

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

$$\sqrt{\left(x-\frac{8}{5}\right)^{2}}=\sqrt{-\frac{81}{25}}$$

Whakarūnātia.

$$x-\frac{8}{5}=\frac{9}{5}i x-\frac{8}{5}=-\frac{9}{5}i$$

Me tāpiri \frac{8}{5} ki ngā taha e rua o te whārite.

$$x=\frac{8}{5}+\frac{9}{5}i x=\frac{8}{5}-\frac{9}{5}i$$