Whakaoti i te n(9n+10)+4=0

$$n ( 9 n + 10 ) + 4 = 0$$

Whakaoti mō n

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te n ki te 9n+10.

$$9n^{2}+10n+4=0$$

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}.

$$n=\frac{-10±\sqrt{10^{2}-4\times 9\times 4}}{2\times 9}$$

Pūrua 10.

$$n=\frac{-10±\sqrt{100-4\times 9\times 4}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$n=\frac{-10±\sqrt{100-36\times 4}}{2\times 9}$$

Whakareatia -36 ki te 4.

$$n=\frac{-10±\sqrt{100-144}}{2\times 9}$$

Tāpiri 100 ki te -144.

$$n=\frac{-10±\sqrt{-44}}{2\times 9}$$

Tuhia te pūtakerua o te -44.

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

Whakareatia 2 ki te 9.

$$n=\frac{-10±2\sqrt{11}i}{18}$$

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

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

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

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

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

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

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

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te n ki te 9n+10.

$$9n^{2}+10n+4=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Whakawehea ngā taha e rua ki te 9.

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

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

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

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.

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

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

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

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.

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

Tauwehea n^{2}+\frac{10}{9}n+\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}.

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

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

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

Whakarūnātia.

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

Me tango \frac{5}{9} mai i ngā taha e rua o te whārite.

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