Whakaoti i te 29x^2+8x+7=0

$$29 x ^ { 2 } + 8 x + 7 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-4+\sqrt{187}i}{29}\approx -0.137931034+0.471544632i$$

$$x=\frac{-\sqrt{187}i-4}{29}\approx -0.137931034-0.471544632i$$

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.

$$29x^{2}+8x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 29 mō a, 8 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 29\times 7}}{2\times 29}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 29\times 7}}{2\times 29}$$

Whakareatia -4 ki te 29.

$$x=\frac{-8±\sqrt{64-116\times 7}}{2\times 29}$$

Whakareatia -116 ki te 7.

$$x=\frac{-8±\sqrt{64-812}}{2\times 29}$$

Tāpiri 64 ki te -812.

$$x=\frac{-8±\sqrt{-748}}{2\times 29}$$

Tuhia te pūtakerua o te -748.

$$x=\frac{-8±2\sqrt{187}i}{2\times 29}$$

Whakareatia 2 ki te 29.

$$x=\frac{-8±2\sqrt{187}i}{58}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{187}i}{58} ina he tāpiri te ±. Tāpiri -8 ki te 2i\sqrt{187}.

$$x=\frac{-8+2\sqrt{187}i}{58}$$

Whakawehe -8+2i\sqrt{187} ki te 58.

$$x=\frac{-4+\sqrt{187}i}{29}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{187}i}{58} ina he tango te ±. Tango 2i\sqrt{187} mai i -8.

$$x=\frac{-2\sqrt{187}i-8}{58}$$

Whakawehe -8-2i\sqrt{187} ki te 58.

$$x=\frac{-\sqrt{187}i-4}{29}$$

Kua oti te whārite te whakatau.

$$x=\frac{-4+\sqrt{187}i}{29} x=\frac{-\sqrt{187}i-4}{29}$$

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.

$$29x^{2}+8x+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$29x^{2}+8x+7-7=-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$29x^{2}+8x=-7$$

Whakawehea ngā taha e rua ki te 29.

$$\frac{29x^{2}+8x}{29}=-\frac{7}{29}$$

Mā te whakawehe ki te 29 ka wetekia te whakareanga ki te 29.

$$x^{2}+\frac{8}{29}x=-\frac{7}{29}$$

Whakawehea te \frac{8}{29}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{4}{29}. Nā, tāpiria te pūrua o te \frac{4}{29} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{8}{29}x+\left(\frac{4}{29}\right)^{2}=-\frac{7}{29}+\left(\frac{4}{29}\right)^{2}$$

Pūruatia \frac{4}{29} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{8}{29}x+\frac{16}{841}=-\frac{7}{29}+\frac{16}{841}$$

Tāpiri -\frac{7}{29} ki te \frac{16}{841} 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.

$$x^{2}+\frac{8}{29}x+\frac{16}{841}=-\frac{187}{841}$$

Tauwehea x^{2}+\frac{8}{29}x+\frac{16}{841}. 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}.

$$\left(x+\frac{4}{29}\right)^{2}=-\frac{187}{841}$$

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

$$\sqrt{\left(x+\frac{4}{29}\right)^{2}}=\sqrt{-\frac{187}{841}}$$

Whakarūnātia.

$$x+\frac{4}{29}=\frac{\sqrt{187}i}{29} x+\frac{4}{29}=-\frac{\sqrt{187}i}{29}$$

Me tango \frac{4}{29} mai i ngā taha e rua o te whārite.

$$x=\frac{-4+\sqrt{187}i}{29} x=\frac{-\sqrt{187}i-4}{29}$$