Whakaoti i te 3z^2+9z+6=0

$$3 { z }^{ 2 } +9z+6=0$$

Whakaoti mō z

$$z=-2$$

$$z=-1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 3.

$$z^{2}+3z+2=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei z^{2}+az+bz+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=1\times 2=2$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=2$$

Tuhia anō te z^{2}+3z+2 hei \left(z^{2}+z\right)+\left(2z+2\right).

$$\left(z^{2}+z\right)+\left(2z+2\right)$$

Tauwehea te z i te tuatahi me te 2 i te rōpū tuarua.

$$z\left(z+1\right)+2\left(z+1\right)$$

Whakatauwehea atu te kīanga pātahi z+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(z+1\right)\left(z+2\right)$$

Hei kimi otinga whārite, me whakaoti te z+1=0 me te z+2=0.

$$z=-1 z=-2$$

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.

$$3z^{2}+9z+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 9 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 9.

$$z=\frac{-9±\sqrt{81-4\times 3\times 6}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$z=\frac{-9±\sqrt{81-12\times 6}}{2\times 3}$$

Whakareatia -12 ki te 6.

$$z=\frac{-9±\sqrt{81-72}}{2\times 3}$$

Tāpiri 81 ki te -72.

$$z=\frac{-9±\sqrt{9}}{2\times 3}$$

Tuhia te pūtakerua o te 9.

$$z=\frac{-9±3}{2\times 3}$$

Whakareatia 2 ki te 3.

$$z=\frac{-9±3}{6}$$

Nā, me whakaoti te whārite z=\frac{-9±3}{6} ina he tāpiri te ±. Tāpiri -9 ki te 3.

$$z=-\frac{6}{6}$$

Whakawehe -6 ki te 6.

$$z=-1$$

Nā, me whakaoti te whārite z=\frac{-9±3}{6} ina he tango te ±. Tango 3 mai i -9.

$$z=-\frac{12}{6}$$

Whakawehe -12 ki te 6.

$$z=-2$$

Kua oti te whārite te whakatau.

$$z=-1 z=-2$$

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.

$$3z^{2}+9z+6=0$$

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

$$3z^{2}+9z+6-6=-6$$

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

$$3z^{2}+9z=-6$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3z^{2}+9z}{3}=-\frac{6}{3}$$

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

$$z^{2}+\frac{9}{3}z=-\frac{6}{3}$$

Whakawehe 9 ki te 3.

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

Whakawehe -6 ki te 3.

$$z^{2}+3z=-2$$

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

$$z^{2}+3z+\left(\frac{3}{2}\right)^{2}=-2+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

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

Tāpiri -2 ki te \frac{9}{4}.

$$z^{2}+3z+\frac{9}{4}=\frac{1}{4}$$

Tauwehea te z^{2}+3z+\frac{9}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(z+\frac{3}{2}\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(z+\frac{3}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$z+\frac{3}{2}=\frac{1}{2} z+\frac{3}{2}=-\frac{1}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$z=-1 z=-2$$