Whakaoti i te 9a^2-10a+4=0

$$9 a ^ { 2 } - 10 a + 4 = 0$$

Whakaoti mō a

$$a=\frac{5+\sqrt{11}i}{9}\approx 0.555555556+0.368513866i$$

$$a=\frac{-\sqrt{11}i+5}{9}\approx 0.555555556-0.368513866i$$

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.

$$9a^{2}-10a+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -10 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 9\times 4}}{2\times 9}$$

Pūrua -10.

$$a=\frac{-\left(-10\right)±\sqrt{100-4\times 9\times 4}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$a=\frac{-\left(-10\right)±\sqrt{100-36\times 4}}{2\times 9}$$

Whakareatia -36 ki te 4.

$$a=\frac{-\left(-10\right)±\sqrt{100-144}}{2\times 9}$$

Tāpiri 100 ki te -144.

$$a=\frac{-\left(-10\right)±\sqrt{-44}}{2\times 9}$$

Tuhia te pūtakerua o te -44.

$$a=\frac{-\left(-10\right)±2\sqrt{11}i}{2\times 9}$$

Ko te tauaro o -10 ko 10.

$$a=\frac{10±2\sqrt{11}i}{2\times 9}$$

Whakareatia 2 ki te 9.

$$a=\frac{10±2\sqrt{11}i}{18}$$

Nā, me whakaoti te whārite a=\frac{10±2\sqrt{11}i}{18} ina he tāpiri te ±. Tāpiri 10 ki te 2i\sqrt{11}.

$$a=\frac{10+2\sqrt{11}i}{18}$$

Whakawehe 10+2i\sqrt{11} ki te 18.

$$a=\frac{5+\sqrt{11}i}{9}$$

Nā, me whakaoti te whārite a=\frac{10±2\sqrt{11}i}{18} ina he tango te ±. Tango 2i\sqrt{11} mai i 10.

$$a=\frac{-2\sqrt{11}i+10}{18}$$

Whakawehe 10-2i\sqrt{11} ki te 18.

$$a=\frac{-\sqrt{11}i+5}{9}$$

Kua oti te whārite te whakatau.

$$a=\frac{5+\sqrt{11}i}{9} a=\frac{-\sqrt{11}i+5}{9}$$

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.

$$9a^{2}-10a+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$9a^{2}-10a+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$9a^{2}-10a=-4$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9a^{2}-10a}{9}=-\frac{4}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$a^{2}-\frac{10}{9}a=-\frac{4}{9}$$

Whakawehea te -\frac{10}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{9}. Nā, tāpiria te pūrua o te -\frac{5}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}-\frac{10}{9}a+\left(-\frac{5}{9}\right)^{2}=-\frac{4}{9}+\left(-\frac{5}{9}\right)^{2}$$

Pūruatia -\frac{5}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$a^{2}-\frac{10}{9}a+\frac{25}{81}=-\frac{4}{9}+\frac{25}{81}$$

Tāpiri -\frac{4}{9} ki te \frac{25}{81} 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.

$$a^{2}-\frac{10}{9}a+\frac{25}{81}=-\frac{11}{81}$$

Tauwehea a^{2}-\frac{10}{9}a+\frac{25}{81}. 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}.

$$\left(a-\frac{5}{9}\right)^{2}=-\frac{11}{81}$$

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

$$\sqrt{\left(a-\frac{5}{9}\right)^{2}}=\sqrt{-\frac{11}{81}}$$

Whakarūnātia.

$$a-\frac{5}{9}=\frac{\sqrt{11}i}{9} a-\frac{5}{9}=-\frac{\sqrt{11}i}{9}$$

Me tāpiri \frac{5}{9} ki ngā taha e rua o te whārite.

$$a=\frac{5+\sqrt{11}i}{9} a=\frac{-\sqrt{11}i+5}{9}$$