Whakaoti i te 56x^2+4x+85=0

$$56 { x }^{ 2 } +4x+85=0$$

Whakaoti mō x (complex solution)

$$x=\frac{-1+\sqrt{1189}i}{28}\approx -0.035714286+1.231495689i$$

$$x=\frac{-\sqrt{1189}i-1}{28}\approx -0.035714286-1.231495689i$$

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.

$$56x^{2}+4x+85=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 56 mō a, 4 mō b, me 85 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 56\times 85}}{2\times 56}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 56\times 85}}{2\times 56}$$

Whakareatia -4 ki te 56.

$$x=\frac{-4±\sqrt{16-224\times 85}}{2\times 56}$$

Whakareatia -224 ki te 85.

$$x=\frac{-4±\sqrt{16-19040}}{2\times 56}$$

Tāpiri 16 ki te -19040.

$$x=\frac{-4±\sqrt{-19024}}{2\times 56}$$

Tuhia te pūtakerua o te -19024.

$$x=\frac{-4±4\sqrt{1189}i}{2\times 56}$$

Whakareatia 2 ki te 56.

$$x=\frac{-4±4\sqrt{1189}i}{112}$$

Nā, me whakaoti te whārite x=\frac{-4±4\sqrt{1189}i}{112} ina he tāpiri te ±. Tāpiri -4 ki te 4i\sqrt{1189}.

$$x=\frac{-4+4\sqrt{1189}i}{112}$$

Whakawehe -4+4i\sqrt{1189} ki te 112.

$$x=\frac{-1+\sqrt{1189}i}{28}$$

Nā, me whakaoti te whārite x=\frac{-4±4\sqrt{1189}i}{112} ina he tango te ±. Tango 4i\sqrt{1189} mai i -4.

$$x=\frac{-4\sqrt{1189}i-4}{112}$$

Whakawehe -4-4i\sqrt{1189} ki te 112.

$$x=\frac{-\sqrt{1189}i-1}{28}$$

Kua oti te whārite te whakatau.

$$x=\frac{-1+\sqrt{1189}i}{28} x=\frac{-\sqrt{1189}i-1}{28}$$

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.

$$56x^{2}+4x+85=0$$

Me tango 85 mai i ngā taha e rua o te whārite.

$$56x^{2}+4x+85-85=-85$$

Mā te tango i te 85 i a ia ake anō ka toe ko te 0.

$$56x^{2}+4x=-85$$

Whakawehea ngā taha e rua ki te 56.

$$\frac{56x^{2}+4x}{56}=-\frac{85}{56}$$

Mā te whakawehe ki te 56 ka wetekia te whakareanga ki te 56.

$$x^{2}+\frac{4}{56}x=-\frac{85}{56}$$

Whakahekea te hautanga \frac{4}{56} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}+\frac{1}{14}x=-\frac{85}{56}$$

Whakawehea te \frac{1}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{28}. Nā, tāpiria te pūrua o te \frac{1}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{14}x+\left(\frac{1}{28}\right)^{2}=-\frac{85}{56}+\left(\frac{1}{28}\right)^{2}$$

Pūruatia \frac{1}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{14}x+\frac{1}{784}=-\frac{85}{56}+\frac{1}{784}$$

Tāpiri -\frac{85}{56} ki te \frac{1}{784} 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.

$$x^{2}+\frac{1}{14}x+\frac{1}{784}=-\frac{1189}{784}$$

Tauwehea x^{2}+\frac{1}{14}x+\frac{1}{784}. 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}.

$$\left(x+\frac{1}{28}\right)^{2}=-\frac{1189}{784}$$

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

$$\sqrt{\left(x+\frac{1}{28}\right)^{2}}=\sqrt{-\frac{1189}{784}}$$

Whakarūnātia.

$$x+\frac{1}{28}=\frac{\sqrt{1189}i}{28} x+\frac{1}{28}=-\frac{\sqrt{1189}i}{28}$$

Me tango \frac{1}{28} mai i ngā taha e rua o te whārite.

$$x=\frac{-1+\sqrt{1189}i}{28} x=\frac{-\sqrt{1189}i-1}{28}$$