Whakaoti i te 1000x^2+2x+69=0

$$1000 { x }^{ 2 } +2x+69=0$$

Whakaoti mō x (complex solution)

$$x=\frac{-1+\sqrt{68999}i}{1000}\approx -0.001+0.262676607i$$

$$x=\frac{-\sqrt{68999}i-1}{1000}\approx -0.001-0.262676607i$$

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.

$$1000x^{2}+2x+69=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1000 mō a, 2 mō b, me 69 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 1000\times 69}}{2\times 1000}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\times 1000\times 69}}{2\times 1000}$$

Whakareatia -4 ki te 1000.

$$x=\frac{-2±\sqrt{4-4000\times 69}}{2\times 1000}$$

Whakareatia -4000 ki te 69.

$$x=\frac{-2±\sqrt{4-276000}}{2\times 1000}$$

Tāpiri 4 ki te -276000.

$$x=\frac{-2±\sqrt{-275996}}{2\times 1000}$$

Tuhia te pūtakerua o te -275996.

$$x=\frac{-2±2\sqrt{68999}i}{2\times 1000}$$

Whakareatia 2 ki te 1000.

$$x=\frac{-2±2\sqrt{68999}i}{2000}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{68999}i}{2000} ina he tāpiri te ±. Tāpiri -2 ki te 2i\sqrt{68999}.

$$x=\frac{-2+2\sqrt{68999}i}{2000}$$

Whakawehe -2+2i\sqrt{68999} ki te 2000.

$$x=\frac{-1+\sqrt{68999}i}{1000}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{68999}i}{2000} ina he tango te ±. Tango 2i\sqrt{68999} mai i -2.

$$x=\frac{-2\sqrt{68999}i-2}{2000}$$

Whakawehe -2-2i\sqrt{68999} ki te 2000.

$$x=\frac{-\sqrt{68999}i-1}{1000}$$

Kua oti te whārite te whakatau.

$$x=\frac{-1+\sqrt{68999}i}{1000} x=\frac{-\sqrt{68999}i-1}{1000}$$

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.

$$1000x^{2}+2x+69=0$$

Me tango 69 mai i ngā taha e rua o te whārite.

$$1000x^{2}+2x+69-69=-69$$

Mā te tango i te 69 i a ia ake anō ka toe ko te 0.

$$1000x^{2}+2x=-69$$

Whakawehea ngā taha e rua ki te 1000.

$$\frac{1000x^{2}+2x}{1000}=-\frac{69}{1000}$$

Mā te whakawehe ki te 1000 ka wetekia te whakareanga ki te 1000.

$$x^{2}+\frac{2}{1000}x=-\frac{69}{1000}$$

Whakahekea te hautanga \frac{2}{1000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{1}{500}x=-\frac{69}{1000}$$

Whakawehea te \frac{1}{500}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{1000}. Nā, tāpiria te pūrua o te \frac{1}{1000} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{500}x+\left(\frac{1}{1000}\right)^{2}=-\frac{69}{1000}+\left(\frac{1}{1000}\right)^{2}$$

Pūruatia \frac{1}{1000} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{500}x+\frac{1}{1000000}=-\frac{69}{1000}+\frac{1}{1000000}$$

Tāpiri -\frac{69}{1000} ki te \frac{1}{1000000} 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.

$$x^{2}+\frac{1}{500}x+\frac{1}{1000000}=-\frac{68999}{1000000}$$

Tauwehea x^{2}+\frac{1}{500}x+\frac{1}{1000000}. 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}.

$$\left(x+\frac{1}{1000}\right)^{2}=-\frac{68999}{1000000}$$

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

$$\sqrt{\left(x+\frac{1}{1000}\right)^{2}}=\sqrt{-\frac{68999}{1000000}}$$

Whakarūnātia.

$$x+\frac{1}{1000}=\frac{\sqrt{68999}i}{1000} x+\frac{1}{1000}=-\frac{\sqrt{68999}i}{1000}$$

Me tango \frac{1}{1000} mai i ngā taha e rua o te whārite.

$$x=\frac{-1+\sqrt{68999}i}{1000} x=\frac{-\sqrt{68999}i-1}{1000}$$