Whakaoti i te 38.706x^2-41.070x+9027=0

$$38.706 x ^ { 2 } - 41.070 x + 9027 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{6845+i\times 5\sqrt{1551010559}}{12902}\approx 0.530537901+15.262312584i$$

$$x=\frac{-i\times 5\sqrt{1551010559}+6845}{12902}\approx 0.530537901-15.262312584i$$

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.

$$38.706x^{2}-41.07x+9027=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 38.706 mō a, -41.07 mō b, me 9027 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-41.07\right)±\sqrt{\left(-41.07\right)^{2}-4\times 38.706\times 9027}}{2\times 38.706}$$

Pūruatia -41.07 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-41.07\right)±\sqrt{1686.7449-4\times 38.706\times 9027}}{2\times 38.706}$$

Whakareatia -4 ki te 38.706.

$$x=\frac{-\left(-41.07\right)±\sqrt{1686.7449-154.824\times 9027}}{2\times 38.706}$$

Whakareatia -154.824 ki te 9027.

$$x=\frac{-\left(-41.07\right)±\sqrt{1686.7449-1397596.248}}{2\times 38.706}$$

Tāpiri 1686.7449 ki te -1397596.248 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.

$$x=\frac{-\left(-41.07\right)±\sqrt{-1395909.5031}}{2\times 38.706}$$

Tuhia te pūtakerua o te -1395909.5031.

$$x=\frac{-\left(-41.07\right)±\frac{3\sqrt{1551010559}i}{100}}{2\times 38.706}$$

Ko te tauaro o -41.07 ko 41.07.

$$x=\frac{41.07±\frac{3\sqrt{1551010559}i}{100}}{2\times 38.706}$$

Whakareatia 2 ki te 38.706.

$$x=\frac{41.07±\frac{3\sqrt{1551010559}i}{100}}{77.412}$$

Nā, me whakaoti te whārite x=\frac{41.07±\frac{3\sqrt{1551010559}i}{100}}{77.412} ina he tāpiri te ±. Tāpiri 41.07 ki te \frac{3i\sqrt{1551010559}}{100}.

$$x=\frac{4107+3\sqrt{1551010559}i}{77.412\times 100}$$

Whakawehe \frac{4107+3i\sqrt{1551010559}}{100} ki te 77.412 mā te whakarea \frac{4107+3i\sqrt{1551010559}}{100} ki te tau huripoki o 77.412.

$$x=\frac{6845+5\sqrt{1551010559}i}{12902}$$

Nā, me whakaoti te whārite x=\frac{41.07±\frac{3\sqrt{1551010559}i}{100}}{77.412} ina he tango te ±. Tango \frac{3i\sqrt{1551010559}}{100} mai i 41.07.

$$x=\frac{-3\sqrt{1551010559}i+4107}{77.412\times 100}$$

Whakawehe \frac{4107-3i\sqrt{1551010559}}{100} ki te 77.412 mā te whakarea \frac{4107-3i\sqrt{1551010559}}{100} ki te tau huripoki o 77.412.

$$x=\frac{-5\sqrt{1551010559}i+6845}{12902}$$

Kua oti te whārite te whakatau.

$$x=\frac{6845+5\sqrt{1551010559}i}{12902} x=\frac{-5\sqrt{1551010559}i+6845}{12902}$$

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.

$$38.706x^{2}-41.07x+9027=0$$

Me tango 9027 mai i ngā taha e rua o te whārite.

$$38.706x^{2}-41.07x+9027-9027=-9027$$

Mā te tango i te 9027 i a ia ake anō ka toe ko te 0.

$$38.706x^{2}-41.07x=-9027$$

Whakawehea ngā taha e rua o te whārite ki te 38.706, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{38.706x^{2}-41.07x}{38.706}=-\frac{9027}{38.706}$$

Mā te whakawehe ki te 38.706 ka wetekia te whakareanga ki te 38.706.

$$x^{2}+\left(-\frac{41.07}{38.706}\right)x=-\frac{9027}{38.706}$$

Whakawehe -41.07 ki te 38.706 mā te whakarea -41.07 ki te tau huripoki o 38.706.

$$x^{2}-\frac{6845}{6451}x=-\frac{9027}{38.706}$$

Whakawehe -9027 ki te 38.706 mā te whakarea -9027 ki te tau huripoki o 38.706.

$$x^{2}-\frac{6845}{6451}x=-\frac{1504500}{6451}$$

Whakawehea te -\frac{6845}{6451}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{6845}{12902}. Nā, tāpiria te pūrua o te -\frac{6845}{12902} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{6845}{6451}x+\left(-\frac{6845}{12902}\right)^{2}=-\frac{1504500}{6451}+\left(-\frac{6845}{12902}\right)^{2}$$

Pūruatia -\frac{6845}{12902} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{6845}{6451}x+\frac{46854025}{166461604}=-\frac{1504500}{6451}+\frac{46854025}{166461604}$$

Tāpiri -\frac{1504500}{6451} ki te \frac{46854025}{166461604} 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.

$$x^{2}-\frac{6845}{6451}x+\frac{46854025}{166461604}=-\frac{38775263975}{166461604}$$

Tauwehea x^{2}-\frac{6845}{6451}x+\frac{46854025}{166461604}. 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}.

$$\left(x-\frac{6845}{12902}\right)^{2}=-\frac{38775263975}{166461604}$$

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

$$\sqrt{\left(x-\frac{6845}{12902}\right)^{2}}=\sqrt{-\frac{38775263975}{166461604}}$$

Whakarūnātia.

$$x-\frac{6845}{12902}=\frac{5\sqrt{1551010559}i}{12902} x-\frac{6845}{12902}=-\frac{5\sqrt{1551010559}i}{12902}$$

Me tāpiri \frac{6845}{12902} ki ngā taha e rua o te whārite.

$$x=\frac{6845+5\sqrt{1551010559}i}{12902} x=\frac{-5\sqrt{1551010559}i+6845}{12902}$$