Whakaoti i te 4m^2+3m+6=0

$$4 m ^ { 2 } + 3 m + 6 = 0$$

Whakaoti mō m

$$m=\frac{-3+\sqrt{87}i}{8}\approx -0.375+1.165922382i$$

$$m=\frac{-\sqrt{87}i-3}{8}\approx -0.375-1.165922382i$$

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.

$$4m^{2}+3m+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 3 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-3±\sqrt{3^{2}-4\times 4\times 6}}{2\times 4}$$

Pūrua 3.

$$m=\frac{-3±\sqrt{9-4\times 4\times 6}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$m=\frac{-3±\sqrt{9-16\times 6}}{2\times 4}$$

Whakareatia -16 ki te 6.

$$m=\frac{-3±\sqrt{9-96}}{2\times 4}$$

Tāpiri 9 ki te -96.

$$m=\frac{-3±\sqrt{-87}}{2\times 4}$$

Tuhia te pūtakerua o te -87.

$$m=\frac{-3±\sqrt{87}i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$m=\frac{-3±\sqrt{87}i}{8}$$

Nā, me whakaoti te whārite m=\frac{-3±\sqrt{87}i}{8} ina he tāpiri te ±. Tāpiri -3 ki te i\sqrt{87}.

$$m=\frac{-3+\sqrt{87}i}{8}$$

Nā, me whakaoti te whārite m=\frac{-3±\sqrt{87}i}{8} ina he tango te ±. Tango i\sqrt{87} mai i -3.

$$m=\frac{-\sqrt{87}i-3}{8}$$

Kua oti te whārite te whakatau.

$$m=\frac{-3+\sqrt{87}i}{8} m=\frac{-\sqrt{87}i-3}{8}$$

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.

$$4m^{2}+3m+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$4m^{2}+3m+6-6=-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$4m^{2}+3m=-6$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4m^{2}+3m}{4}=-\frac{6}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$m^{2}+\frac{3}{4}m=-\frac{6}{4}$$

Whakahekea te hautanga \frac{-6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$m^{2}+\frac{3}{4}m=-\frac{3}{2}$$

Whakawehea te \frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{8}. Nā, tāpiria te pūrua o te \frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$m^{2}+\frac{3}{4}m+\left(\frac{3}{8}\right)^{2}=-\frac{3}{2}+\left(\frac{3}{8}\right)^{2}$$

Pūruatia \frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$m^{2}+\frac{3}{4}m+\frac{9}{64}=-\frac{3}{2}+\frac{9}{64}$$

Tāpiri -\frac{3}{2} ki te \frac{9}{64} 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.

$$m^{2}+\frac{3}{4}m+\frac{9}{64}=-\frac{87}{64}$$

Tauwehea m^{2}+\frac{3}{4}m+\frac{9}{64}. 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}.

$$\left(m+\frac{3}{8}\right)^{2}=-\frac{87}{64}$$

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

$$\sqrt{\left(m+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{87}{64}}$$

Whakarūnātia.

$$m+\frac{3}{8}=\frac{\sqrt{87}i}{8} m+\frac{3}{8}=-\frac{\sqrt{87}i}{8}$$

Me tango \frac{3}{8} mai i ngā taha e rua o te whārite.

$$m=\frac{-3+\sqrt{87}i}{8} m=\frac{-\sqrt{87}i-3}{8}$$