Whakaoti i te 3y^2+7y+6=0

$$3 y ^ { 2 } + 7 y + 6 = 0$$

Whakaoti mō y

$$y=\frac{-7+\sqrt{23}i}{6}\approx -1.166666667+0.799305254i$$

$$y=\frac{-\sqrt{23}i-7}{6}\approx -1.166666667-0.799305254i$$

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.

$$3y^{2}+7y+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 7 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-7±\sqrt{7^{2}-4\times 3\times 6}}{2\times 3}$$

Pūrua 7.

$$y=\frac{-7±\sqrt{49-4\times 3\times 6}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$y=\frac{-7±\sqrt{49-12\times 6}}{2\times 3}$$

Whakareatia -12 ki te 6.

$$y=\frac{-7±\sqrt{49-72}}{2\times 3}$$

Tāpiri 49 ki te -72.

$$y=\frac{-7±\sqrt{-23}}{2\times 3}$$

Tuhia te pūtakerua o te -23.

$$y=\frac{-7±\sqrt{23}i}{2\times 3}$$

Whakareatia 2 ki te 3.

$$y=\frac{-7±\sqrt{23}i}{6}$$

Nā, me whakaoti te whārite y=\frac{-7±\sqrt{23}i}{6} ina he tāpiri te ±. Tāpiri -7 ki te i\sqrt{23}.

$$y=\frac{-7+\sqrt{23}i}{6}$$

Nā, me whakaoti te whārite y=\frac{-7±\sqrt{23}i}{6} ina he tango te ±. Tango i\sqrt{23} mai i -7.

$$y=\frac{-\sqrt{23}i-7}{6}$$

Kua oti te whārite te whakatau.

$$y=\frac{-7+\sqrt{23}i}{6} y=\frac{-\sqrt{23}i-7}{6}$$

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.

$$3y^{2}+7y+6=0$$

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

$$3y^{2}+7y+6-6=-6$$

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

$$3y^{2}+7y=-6$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3y^{2}+7y}{3}=-\frac{6}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$y^{2}+\frac{7}{3}y=-\frac{6}{3}$$

Whakawehe -6 ki te 3.

$$y^{2}+\frac{7}{3}y=-2$$

Whakawehea te \frac{7}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{6}. Nā, tāpiria te pūrua o te \frac{7}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+\frac{7}{3}y+\left(\frac{7}{6}\right)^{2}=-2+\left(\frac{7}{6}\right)^{2}$$

Pūruatia \frac{7}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}+\frac{7}{3}y+\frac{49}{36}=-2+\frac{49}{36}$$

Tāpiri -2 ki te \frac{49}{36}.

$$y^{2}+\frac{7}{3}y+\frac{49}{36}=-\frac{23}{36}$$

Tauwehea y^{2}+\frac{7}{3}y+\frac{49}{36}. 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}.

$$\left(y+\frac{7}{6}\right)^{2}=-\frac{23}{36}$$

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

$$\sqrt{\left(y+\frac{7}{6}\right)^{2}}=\sqrt{-\frac{23}{36}}$$

Whakarūnātia.

$$y+\frac{7}{6}=\frac{\sqrt{23}i}{6} y+\frac{7}{6}=-\frac{\sqrt{23}i}{6}$$

Me tango \frac{7}{6} mai i ngā taha e rua o te whārite.

$$y=\frac{-7+\sqrt{23}i}{6} y=\frac{-\sqrt{23}i-7}{6}$$