Whakaoti i te x^2(6%)^2+(1-x)^2(2%)^2+2x(1-x)*012*6%*2%=00327

$$x ^ { 2 } ( 6 \% ) ^ { 2 } + ( 1 - x ) ^ { 2 } ( 2 \% ) ^ { 2 } + 2 x ( 1 - x ) \times 012 \times 6 \% \times 2 \% = 00327$$

Whakaoti mō x (complex solution)

$$x=\frac{1}{10}+\frac{3}{10}i=0.1+0.3i$$

$$x=\frac{1}{10}-\frac{3}{10}i=0.1-0.3i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakahekea te hautanga \frac{6}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}\times \left(\frac{3}{50}\right)^{2}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Tātaihia te \frac{3}{50} mā te pū o 2, kia riro ko \frac{9}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(1-x\right)^{2}.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakahekea te hautanga \frac{2}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{1}{50}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Tātaihia te \frac{1}{50} mā te pū o 2, kia riro ko \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \frac{1}{2500}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1-2x+x^{2} ki te \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\frac{1}{2500}-\frac{1}{1250}x+\frac{1}{2500}x^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Pahekotia te x^{2}\times \frac{9}{2500} me \frac{1}{2500}x^{2}, ka \frac{1}{250}x^{2}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakareatia te 2 ki te 0, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakareatia te 0 ki te 12, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakahekea te hautanga \frac{6}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{3}{50}\times \frac{2}{100}=0\times 0\times 327$$

Whakareatia te 0 ki te \frac{3}{50}, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{2}{100}=0\times 0\times 327$$

Whakahekea te hautanga \frac{2}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{1}{50}=0\times 0\times 327$$

Whakareatia te 0 ki te \frac{1}{50}, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)=0\times 0\times 327$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0=0\times 0\times 327$$

Tāpirihia te \frac{1}{2500} ki te 0, ka \frac{1}{2500}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x=0\times 0\times 327$$

Whakareatia te 0 ki te 0, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x=0\times 327$$

Whakareatia te 0 ki te 327, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x=0$$

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.

$$\frac{1}{250}x^{2}-\frac{1}{1250}x+\frac{1}{2500}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{250} mō a, -\frac{1}{1250} mō b, me \frac{1}{2500} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{1250}\right)±\sqrt{\left(-\frac{1}{1250}\right)^{2}-4\times \frac{1}{250}\times \frac{1}{2500}}}{2\times \frac{1}{250}}$$

Pūruatia -\frac{1}{1250} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{1}{1250}\right)±\sqrt{\frac{1}{1562500}-4\times \frac{1}{250}\times \frac{1}{2500}}}{2\times \frac{1}{250}}$$

Whakareatia -4 ki te \frac{1}{250}.

$$x=\frac{-\left(-\frac{1}{1250}\right)±\sqrt{\frac{1}{1562500}-\frac{2}{125}\times \frac{1}{2500}}}{2\times \frac{1}{250}}$$

Whakareatia -\frac{2}{125} ki te \frac{1}{2500} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\left(-\frac{1}{1250}\right)±\sqrt{\frac{1}{1562500}-\frac{1}{156250}}}{2\times \frac{1}{250}}$$

Tāpiri \frac{1}{1562500} ki te -\frac{1}{156250} 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.

$$x=\frac{-\left(-\frac{1}{1250}\right)±\sqrt{-\frac{9}{1562500}}}{2\times \frac{1}{250}}$$

Tuhia te pūtakerua o te -\frac{9}{1562500}.

$$x=\frac{-\left(-\frac{1}{1250}\right)±\frac{3}{1250}i}{2\times \frac{1}{250}}$$

Ko te tauaro o -\frac{1}{1250} ko \frac{1}{1250}.

$$x=\frac{\frac{1}{1250}±\frac{3}{1250}i}{2\times \frac{1}{250}}$$

Whakareatia 2 ki te \frac{1}{250}.

$$x=\frac{\frac{1}{1250}±\frac{3}{1250}i}{\frac{1}{125}}$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{1250}±\frac{3}{1250}i}{\frac{1}{125}} ina he tāpiri te ±. Tāpiri \frac{1}{1250} ki te \frac{3}{1250}i.

$$x=\frac{\frac{1}{1250}+\frac{3}{1250}i}{\frac{1}{125}}$$

Whakawehe \frac{1}{1250}+\frac{3}{1250}i ki te \frac{1}{125} mā te whakarea \frac{1}{1250}+\frac{3}{1250}i ki te tau huripoki o \frac{1}{125}.

$$x=\frac{1}{10}+\frac{3}{10}i$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{1250}±\frac{3}{1250}i}{\frac{1}{125}} ina he tango te ±. Tango \frac{3}{1250}i mai i \frac{1}{1250}.

$$x=\frac{\frac{1}{1250}-\frac{3}{1250}i}{\frac{1}{125}}$$

Whakawehe \frac{1}{1250}-\frac{3}{1250}i ki te \frac{1}{125} mā te whakarea \frac{1}{1250}-\frac{3}{1250}i ki te tau huripoki o \frac{1}{125}.

$$x=\frac{1}{10}-\frac{3}{10}i$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{10}+\frac{3}{10}i x=\frac{1}{10}-\frac{3}{10}i$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakahekea te hautanga \frac{6}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}\times \left(\frac{3}{50}\right)^{2}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Tātaihia te \frac{3}{50} mā te pū o 2, kia riro ko \frac{9}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-x\right)^{2}\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(1-x\right)^{2}.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{2}{100}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakahekea te hautanga \frac{2}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \left(\frac{1}{50}\right)^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Tātaihia te \frac{1}{50} mā te pū o 2, kia riro ko \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\left(1-2x+x^{2}\right)\times \frac{1}{2500}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1-2x+x^{2} ki te \frac{1}{2500}.

$$x^{2}\times \frac{9}{2500}+\frac{1}{2500}-\frac{1}{1250}x+\frac{1}{2500}x^{2}+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Pahekotia te x^{2}\times \frac{9}{2500} me \frac{1}{2500}x^{2}, ka \frac{1}{250}x^{2}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+2x\left(1-x\right)\times 0\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakareatia te 2 ki te 0, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times 12\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakareatia te 0 ki te 12, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{6}{100}\times \frac{2}{100}=0\times 0\times 327$$

Whakahekea te hautanga \frac{6}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{3}{50}\times \frac{2}{100}=0\times 0\times 327$$

Whakareatia te 0 ki te \frac{3}{50}, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{2}{100}=0\times 0\times 327$$

Whakahekea te hautanga \frac{2}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)\times \frac{1}{50}=0\times 0\times 327$$

Whakareatia te 0 ki te \frac{1}{50}, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0x\left(1-x\right)=0\times 0\times 327$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x+0=0\times 0\times 327$$

Tāpirihia te \frac{1}{2500} ki te 0, ka \frac{1}{2500}.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x=0\times 0\times 327$$

Whakareatia te 0 ki te 0, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x=0\times 327$$

Whakareatia te 0 ki te 327, ka 0.

$$\frac{1}{250}x^{2}+\frac{1}{2500}-\frac{1}{1250}x=0$$

Tangohia te \frac{1}{2500} mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$\frac{1}{250}x^{2}-\frac{1}{1250}x=-\frac{1}{2500}$$

Me whakarea ngā taha e rua ki te 250.

$$\frac{\frac{1}{250}x^{2}-\frac{1}{1250}x}{\frac{1}{250}}=-\frac{\frac{1}{2500}}{\frac{1}{250}}$$

Mā te whakawehe ki te \frac{1}{250} ka wetekia te whakareanga ki te \frac{1}{250}.

$$x^{2}+\left(-\frac{\frac{1}{1250}}{\frac{1}{250}}\right)x=-\frac{\frac{1}{2500}}{\frac{1}{250}}$$

Whakawehe -\frac{1}{1250} ki te \frac{1}{250} mā te whakarea -\frac{1}{1250} ki te tau huripoki o \frac{1}{250}.

$$x^{2}-\frac{1}{5}x=-\frac{\frac{1}{2500}}{\frac{1}{250}}$$

Whakawehe -\frac{1}{2500} ki te \frac{1}{250} mā te whakarea -\frac{1}{2500} ki te tau huripoki o \frac{1}{250}.

$$x^{2}-\frac{1}{5}x=-\frac{1}{10}$$

Whakawehea te -\frac{1}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{10}. Nā, tāpiria te pūrua o te -\frac{1}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{5}x+\left(-\frac{1}{10}\right)^{2}=-\frac{1}{10}+\left(-\frac{1}{10}\right)^{2}$$

Pūruatia -\frac{1}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=-\frac{1}{10}+\frac{1}{100}$$

Tāpiri -\frac{1}{10} ki te \frac{1}{100} 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.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=-\frac{9}{100}$$

Tauwehea x^{2}-\frac{1}{5}x+\frac{1}{100}. 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}.

$$\left(x-\frac{1}{10}\right)^{2}=-\frac{9}{100}$$

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

$$\sqrt{\left(x-\frac{1}{10}\right)^{2}}=\sqrt{-\frac{9}{100}}$$

Whakarūnātia.

$$x-\frac{1}{10}=\frac{3}{10}i x-\frac{1}{10}=-\frac{3}{10}i$$

Me tāpiri \frac{1}{10} ki ngā taha e rua o te whārite.

$$x=\frac{1}{10}+\frac{3}{10}i x=\frac{1}{10}-\frac{3}{10}i$$