Whakaoti i te 1024x^2+768x+1280=0

$$1024 { x }^{ 2 } +768x+1280=0$$

Whakaoti mō x (complex solution)

$$x=\frac{-3+\sqrt{71}i}{8}\approx -0.375+1.053268722i$$

$$x=\frac{-\sqrt{71}i-3}{8}\approx -0.375-1.053268722i$$

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.

$$1024x^{2}+768x+1280=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1024 mō a, 768 mō b, me 1280 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-768±\sqrt{768^{2}-4\times 1024\times 1280}}{2\times 1024}$$

Pūrua 768.

$$x=\frac{-768±\sqrt{589824-4\times 1024\times 1280}}{2\times 1024}$$

Whakareatia -4 ki te 1024.

$$x=\frac{-768±\sqrt{589824-4096\times 1280}}{2\times 1024}$$

Whakareatia -4096 ki te 1280.

$$x=\frac{-768±\sqrt{589824-5242880}}{2\times 1024}$$

Tāpiri 589824 ki te -5242880.

$$x=\frac{-768±\sqrt{-4653056}}{2\times 1024}$$

Tuhia te pūtakerua o te -4653056.

$$x=\frac{-768±256\sqrt{71}i}{2\times 1024}$$

Whakareatia 2 ki te 1024.

$$x=\frac{-768±256\sqrt{71}i}{2048}$$

Nā, me whakaoti te whārite x=\frac{-768±256\sqrt{71}i}{2048} ina he tāpiri te ±. Tāpiri -768 ki te 256i\sqrt{71}.

$$x=\frac{-768+256\sqrt{71}i}{2048}$$

Whakawehe -768+256i\sqrt{71} ki te 2048.

$$x=\frac{-3+\sqrt{71}i}{8}$$

Nā, me whakaoti te whārite x=\frac{-768±256\sqrt{71}i}{2048} ina he tango te ±. Tango 256i\sqrt{71} mai i -768.

$$x=\frac{-256\sqrt{71}i-768}{2048}$$

Whakawehe -768-256i\sqrt{71} ki te 2048.

$$x=\frac{-\sqrt{71}i-3}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-3}{8}$$

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.

$$1024x^{2}+768x+1280=0$$

Me tango 1280 mai i ngā taha e rua o te whārite.

$$1024x^{2}+768x+1280-1280=-1280$$

Mā te tango i te 1280 i a ia ake anō ka toe ko te 0.

$$1024x^{2}+768x=-1280$$

Whakawehea ngā taha e rua ki te 1024.

$$\frac{1024x^{2}+768x}{1024}=-\frac{1280}{1024}$$

Mā te whakawehe ki te 1024 ka wetekia te whakareanga ki te 1024.

$$x^{2}+\frac{768}{1024}x=-\frac{1280}{1024}$$

Whakahekea te hautanga \frac{768}{1024} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 256.

$$x^{2}+\frac{3}{4}x=-\frac{1280}{1024}$$

Whakahekea te hautanga \frac{-1280}{1024} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 256.

$$x^{2}+\frac{3}{4}x=-\frac{5}{4}$$

Whakawehea te \frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{8}. Nā, tāpiria te pūrua o te \frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{5}{4}+\left(\frac{3}{8}\right)^{2}$$

Pūruatia \frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{5}{4}+\frac{9}{64}$$

Tāpiri -\frac{5}{4} ki te \frac{9}{64} 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.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{71}{64}$$

Tauwehea x^{2}+\frac{3}{4}x+\frac{9}{64}. 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}.

$$\left(x+\frac{3}{8}\right)^{2}=-\frac{71}{64}$$

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

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{71}{64}}$$

Whakarūnātia.

$$x+\frac{3}{8}=\frac{\sqrt{71}i}{8} x+\frac{3}{8}=-\frac{\sqrt{71}i}{8}$$

Me tango \frac{3}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{-3+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-3}{8}$$