Whakaoti i te 1044*10^-3p=83145*29815(1-186*10^-6p+106*10^-8p^2)

$$1044 \times 10 ^ { - 3 } p = 83145 \times 29815 ( 1 - 186 \times 10 ^ { - 6 } p + 106 \times 10 ^ { - 8 } p ^ { 2 } )$$

Whakaoti mō p

$$p=\frac{153696374850+50\sqrt{1148595450707205174991}i}{1751804177}\approx 87.736047709+967.315156682i$$

$$p=\frac{-50\sqrt{1148595450707205174991}i+153696374850}{1751804177}\approx 87.736047709-967.315156682i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 10 mā te pū o -3, kia riro ko \frac{1}{1000}.

$$1044\times \frac{1}{1000}p=83145\times 29815\left(1-186\times 10^{-6}p+106\times 10^{-8}p^{2}\right)$$

Whakareatia te 1044 ki te \frac{1}{1000}, ka \frac{261}{250}.

$$\frac{261}{250}p=83145\times 29815\left(1-186\times 10^{-6}p+106\times 10^{-8}p^{2}\right)$$

Whakareatia te 83145 ki te 29815, ka 2478968175.

$$\frac{261}{250}p=2478968175\left(1-186\times 10^{-6}p+106\times 10^{-8}p^{2}\right)$$

Tātaihia te 10 mā te pū o -6, kia riro ko \frac{1}{1000000}.

$$\frac{261}{250}p=2478968175\left(1-186\times \frac{1}{1000000}p+106\times 10^{-8}p^{2}\right)$$

Whakareatia te 186 ki te \frac{1}{1000000}, ka \frac{93}{500000}.

$$\frac{261}{250}p=2478968175\left(1-\frac{93}{500000}p+106\times 10^{-8}p^{2}\right)$$

Tātaihia te 10 mā te pū o -8, kia riro ko \frac{1}{100000000}.

$$\frac{261}{250}p=2478968175\left(1-\frac{93}{500000}p+106\times \frac{1}{100000000}p^{2}\right)$$

Whakareatia te 106 ki te \frac{1}{100000000}, ka \frac{53}{50000000}.

$$\frac{261}{250}p=2478968175\left(1-\frac{93}{500000}p+\frac{53}{50000000}p^{2}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2478968175 ki te 1-\frac{93}{500000}p+\frac{53}{50000000}p^{2}.

$$\frac{261}{250}p=2478968175-\frac{9221761611}{20000}p+\frac{5255412531}{2000000}p^{2}$$

Tangohia te 2478968175 mai i ngā taha e rua.

$$\frac{261}{250}p-2478968175=-\frac{9221761611}{20000}p+\frac{5255412531}{2000000}p^{2}$$

Me tāpiri te \frac{9221761611}{20000}p ki ngā taha e rua.

$$\frac{261}{250}p-2478968175+\frac{9221761611}{20000}p=\frac{5255412531}{2000000}p^{2}$$

Pahekotia te \frac{261}{250}p me \frac{9221761611}{20000}p, ka \frac{9221782491}{20000}p.

$$\frac{9221782491}{20000}p-2478968175=\frac{5255412531}{2000000}p^{2}$$

Tangohia te \frac{5255412531}{2000000}p^{2} mai i ngā taha e rua.

$$\frac{9221782491}{20000}p-2478968175-\frac{5255412531}{2000000}p^{2}=0$$

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.

$$-\frac{5255412531}{2000000}p^{2}+\frac{9221782491}{20000}p-2478968175=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{5255412531}{2000000} mō a, \frac{9221782491}{20000} mō b, me -2478968175 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\frac{9221782491}{20000}±\sqrt{\left(\frac{9221782491}{20000}\right)^{2}-4\left(-\frac{5255412531}{2000000}\right)\left(-2478968175\right)}}{2\left(-\frac{5255412531}{2000000}\right)}$$

Pūruatia \frac{9221782491}{20000} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$p=\frac{-\frac{9221782491}{20000}±\sqrt{\frac{85041272311314165081}{400000000}-4\left(-\frac{5255412531}{2000000}\right)\left(-2478968175\right)}}{2\left(-\frac{5255412531}{2000000}\right)}$$

Whakareatia -4 ki te -\frac{5255412531}{2000000}.

$$p=\frac{-\frac{9221782491}{20000}±\sqrt{\frac{85041272311314165081}{400000000}+\frac{5255412531}{500000}\left(-2478968175\right)}}{2\left(-\frac{5255412531}{2000000}\right)}$$

Whakareatia \frac{5255412531}{500000} ki te -2478968175.

$$p=\frac{-\frac{9221782491}{20000}±\sqrt{\frac{85041272311314165081}{400000000}-\frac{521120016433808037}{20000}}}{2\left(-\frac{5255412531}{2000000}\right)}$$

Tāpiri \frac{85041272311314165081}{400000000} ki te -\frac{521120016433808037}{20000} 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.

$$p=\frac{-\frac{9221782491}{20000}±\sqrt{-\frac{10337359056364846574919}{400000000}}}{2\left(-\frac{5255412531}{2000000}\right)}$$

Tuhia te pūtakerua o te -\frac{10337359056364846574919}{400000000}.

$$p=\frac{-\frac{9221782491}{20000}±\frac{3\sqrt{1148595450707205174991}i}{20000}}{2\left(-\frac{5255412531}{2000000}\right)}$$

Whakareatia 2 ki te -\frac{5255412531}{2000000}.

$$p=\frac{-\frac{9221782491}{20000}±\frac{3\sqrt{1148595450707205174991}i}{20000}}{-\frac{5255412531}{1000000}}$$

Nā, me whakaoti te whārite p=\frac{-\frac{9221782491}{20000}±\frac{3\sqrt{1148595450707205174991}i}{20000}}{-\frac{5255412531}{1000000}} ina he tāpiri te ±. Tāpiri -\frac{9221782491}{20000} ki te \frac{3i\sqrt{1148595450707205174991}}{20000}.

$$p=\frac{-9221782491+3\sqrt{1148595450707205174991}i}{-\frac{5255412531}{1000000}\times 20000}$$

Whakawehe \frac{-9221782491+3i\sqrt{1148595450707205174991}}{20000} ki te -\frac{5255412531}{1000000} mā te whakarea \frac{-9221782491+3i\sqrt{1148595450707205174991}}{20000} ki te tau huripoki o -\frac{5255412531}{1000000}.

$$p=\frac{-50\sqrt{1148595450707205174991}i+153696374850}{1751804177}$$

Nā, me whakaoti te whārite p=\frac{-\frac{9221782491}{20000}±\frac{3\sqrt{1148595450707205174991}i}{20000}}{-\frac{5255412531}{1000000}} ina he tango te ±. Tango \frac{3i\sqrt{1148595450707205174991}}{20000} mai i -\frac{9221782491}{20000}.

$$p=\frac{-3\sqrt{1148595450707205174991}i-9221782491}{-\frac{5255412531}{1000000}\times 20000}$$

Whakawehe \frac{-9221782491-3i\sqrt{1148595450707205174991}}{20000} ki te -\frac{5255412531}{1000000} mā te whakarea \frac{-9221782491-3i\sqrt{1148595450707205174991}}{20000} ki te tau huripoki o -\frac{5255412531}{1000000}.

$$p=\frac{153696374850+50\sqrt{1148595450707205174991}i}{1751804177}$$

Kua oti te whārite te whakatau.

$$p=\frac{-50\sqrt{1148595450707205174991}i+153696374850}{1751804177} p=\frac{153696374850+50\sqrt{1148595450707205174991}i}{1751804177}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 10 mā te pū o -3, kia riro ko \frac{1}{1000}.

$$1044\times \frac{1}{1000}p=83145\times 29815\left(1-186\times 10^{-6}p+106\times 10^{-8}p^{2}\right)$$

Whakareatia te 1044 ki te \frac{1}{1000}, ka \frac{261}{250}.

$$\frac{261}{250}p=83145\times 29815\left(1-186\times 10^{-6}p+106\times 10^{-8}p^{2}\right)$$

Whakareatia te 83145 ki te 29815, ka 2478968175.

$$\frac{261}{250}p=2478968175\left(1-186\times 10^{-6}p+106\times 10^{-8}p^{2}\right)$$

Tātaihia te 10 mā te pū o -6, kia riro ko \frac{1}{1000000}.

$$\frac{261}{250}p=2478968175\left(1-186\times \frac{1}{1000000}p+106\times 10^{-8}p^{2}\right)$$

Whakareatia te 186 ki te \frac{1}{1000000}, ka \frac{93}{500000}.

$$\frac{261}{250}p=2478968175\left(1-\frac{93}{500000}p+106\times 10^{-8}p^{2}\right)$$

Tātaihia te 10 mā te pū o -8, kia riro ko \frac{1}{100000000}.

$$\frac{261}{250}p=2478968175\left(1-\frac{93}{500000}p+106\times \frac{1}{100000000}p^{2}\right)$$

Whakareatia te 106 ki te \frac{1}{100000000}, ka \frac{53}{50000000}.

$$\frac{261}{250}p=2478968175\left(1-\frac{93}{500000}p+\frac{53}{50000000}p^{2}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2478968175 ki te 1-\frac{93}{500000}p+\frac{53}{50000000}p^{2}.

$$\frac{261}{250}p=2478968175-\frac{9221761611}{20000}p+\frac{5255412531}{2000000}p^{2}$$

Me tāpiri te \frac{9221761611}{20000}p ki ngā taha e rua.

$$\frac{261}{250}p+\frac{9221761611}{20000}p=2478968175+\frac{5255412531}{2000000}p^{2}$$

Pahekotia te \frac{261}{250}p me \frac{9221761611}{20000}p, ka \frac{9221782491}{20000}p.

$$\frac{9221782491}{20000}p=2478968175+\frac{5255412531}{2000000}p^{2}$$

Tangohia te \frac{5255412531}{2000000}p^{2} mai i ngā taha e rua.

$$\frac{9221782491}{20000}p-\frac{5255412531}{2000000}p^{2}=2478968175$$

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.

$$-\frac{5255412531}{2000000}p^{2}+\frac{9221782491}{20000}p=2478968175$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5255412531}{2000000}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{5255412531}{2000000}p^{2}+\frac{9221782491}{20000}p}{-\frac{5255412531}{2000000}}=\frac{2478968175}{-\frac{5255412531}{2000000}}$$

Mā te whakawehe ki te -\frac{5255412531}{2000000} ka wetekia te whakareanga ki te -\frac{5255412531}{2000000}.

$$p^{2}+\frac{\frac{9221782491}{20000}}{-\frac{5255412531}{2000000}}p=\frac{2478968175}{-\frac{5255412531}{2000000}}$$

Whakawehe \frac{9221782491}{20000} ki te -\frac{5255412531}{2000000} mā te whakarea \frac{9221782491}{20000} ki te tau huripoki o -\frac{5255412531}{2000000}.

$$p^{2}-\frac{307392749700}{1751804177}p=\frac{2478968175}{-\frac{5255412531}{2000000}}$$

Whakawehe 2478968175 ki te -\frac{5255412531}{2000000} mā te whakarea 2478968175 ki te tau huripoki o -\frac{5255412531}{2000000}.

$$p^{2}-\frac{307392749700}{1751804177}p=-\frac{50000000}{53}$$

Whakawehea te -\frac{307392749700}{1751804177}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{153696374850}{1751804177}. Nā, tāpiria te pūrua o te -\frac{153696374850}{1751804177} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$p^{2}-\frac{307392749700}{1751804177}p+\left(-\frac{153696374850}{1751804177}\right)^{2}=-\frac{50000000}{53}+\left(-\frac{153696374850}{1751804177}\right)^{2}$$

Pūruatia -\frac{153696374850}{1751804177} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$p^{2}-\frac{307392749700}{1751804177}p+\frac{23622575642031712522500}{3068817874554647329}=-\frac{50000000}{53}+\frac{23622575642031712522500}{3068817874554647329}$$

Tāpiri -\frac{50000000}{53} ki te \frac{23622575642031712522500}{3068817874554647329} 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.

$$p^{2}-\frac{307392749700}{1751804177}p+\frac{23622575642031712522500}{3068817874554647329}=-\frac{2871488626768012937477500}{3068817874554647329}$$

Tauwehea p^{2}-\frac{307392749700}{1751804177}p+\frac{23622575642031712522500}{3068817874554647329}. 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}.

$$\left(p-\frac{153696374850}{1751804177}\right)^{2}=-\frac{2871488626768012937477500}{3068817874554647329}$$

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

$$\sqrt{\left(p-\frac{153696374850}{1751804177}\right)^{2}}=\sqrt{-\frac{2871488626768012937477500}{3068817874554647329}}$$

Whakarūnātia.

$$p-\frac{153696374850}{1751804177}=\frac{50\sqrt{1148595450707205174991}i}{1751804177} p-\frac{153696374850}{1751804177}=-\frac{50\sqrt{1148595450707205174991}i}{1751804177}$$

Me tāpiri \frac{153696374850}{1751804177} ki ngā taha e rua o te whārite.

$$p=\frac{153696374850+50\sqrt{1148595450707205174991}i}{1751804177} p=\frac{-50\sqrt{1148595450707205174991}i+153696374850}{1751804177}$$