Whakaoti i te 1000x^2+6125x+125=0

$$1000 { x }^{ 2 } +6125x+125=0$$

Whakaoti mō x

$$x=\frac{\sqrt{2369}-49}{16}\approx -0.020476619$$

$$x=\frac{-\sqrt{2369}-49}{16}\approx -6.104523381$$

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.

$$1000x^{2}+6125x+125=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1000 mō a, 6125 mō b, me 125 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6125±\sqrt{6125^{2}-4\times 1000\times 125}}{2\times 1000}$$

Pūrua 6125.

$$x=\frac{-6125±\sqrt{37515625-4\times 1000\times 125}}{2\times 1000}$$

Whakareatia -4 ki te 1000.

$$x=\frac{-6125±\sqrt{37515625-4000\times 125}}{2\times 1000}$$

Whakareatia -4000 ki te 125.

$$x=\frac{-6125±\sqrt{37515625-500000}}{2\times 1000}$$

Tāpiri 37515625 ki te -500000.

$$x=\frac{-6125±\sqrt{37015625}}{2\times 1000}$$

Tuhia te pūtakerua o te 37015625.

$$x=\frac{-6125±125\sqrt{2369}}{2\times 1000}$$

Whakareatia 2 ki te 1000.

$$x=\frac{-6125±125\sqrt{2369}}{2000}$$

Nā, me whakaoti te whārite x=\frac{-6125±125\sqrt{2369}}{2000} ina he tāpiri te ±. Tāpiri -6125 ki te 125\sqrt{2369}.

$$x=\frac{125\sqrt{2369}-6125}{2000}$$

Whakawehe -6125+125\sqrt{2369} ki te 2000.

$$x=\frac{\sqrt{2369}-49}{16}$$

Nā, me whakaoti te whārite x=\frac{-6125±125\sqrt{2369}}{2000} ina he tango te ±. Tango 125\sqrt{2369} mai i -6125.

$$x=\frac{-125\sqrt{2369}-6125}{2000}$$

Whakawehe -6125-125\sqrt{2369} ki te 2000.

$$x=\frac{-\sqrt{2369}-49}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{2369}-49}{16} x=\frac{-\sqrt{2369}-49}{16}$$

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.

$$1000x^{2}+6125x+125=0$$

Me tango 125 mai i ngā taha e rua o te whārite.

$$1000x^{2}+6125x+125-125=-125$$

Mā te tango i te 125 i a ia ake anō ka toe ko te 0.

$$1000x^{2}+6125x=-125$$

Whakawehea ngā taha e rua ki te 1000.

$$\frac{1000x^{2}+6125x}{1000}=-\frac{125}{1000}$$

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

$$x^{2}+\frac{6125}{1000}x=-\frac{125}{1000}$$

Whakahekea te hautanga \frac{6125}{1000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 125.

$$x^{2}+\frac{49}{8}x=-\frac{125}{1000}$$

Whakahekea te hautanga \frac{-125}{1000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 125.

$$x^{2}+\frac{49}{8}x=-\frac{1}{8}$$

Whakawehea te \frac{49}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{49}{16}. Nā, tāpiria te pūrua o te \frac{49}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{49}{8}x+\left(\frac{49}{16}\right)^{2}=-\frac{1}{8}+\left(\frac{49}{16}\right)^{2}$$

Pūruatia \frac{49}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{49}{8}x+\frac{2401}{256}=-\frac{1}{8}+\frac{2401}{256}$$

Tāpiri -\frac{1}{8} ki te \frac{2401}{256} 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.

$$x^{2}+\frac{49}{8}x+\frac{2401}{256}=\frac{2369}{256}$$

Tauwehea x^{2}+\frac{49}{8}x+\frac{2401}{256}. 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}.

$$\left(x+\frac{49}{16}\right)^{2}=\frac{2369}{256}$$

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

$$\sqrt{\left(x+\frac{49}{16}\right)^{2}}=\sqrt{\frac{2369}{256}}$$

Whakarūnātia.

$$x+\frac{49}{16}=\frac{\sqrt{2369}}{16} x+\frac{49}{16}=-\frac{\sqrt{2369}}{16}$$

Me tango \frac{49}{16} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{2369}-49}{16} x=\frac{-\sqrt{2369}-49}{16}$$