Whakaoti i te 3w^2+6=-11w

$$3 w ^ { 2 } + 6 = - 11 w$$

Whakaoti mō w

$$w=-3$$

$$w=-\frac{2}{3}\approx -0.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 11w ki ngā taha e rua.

$$3w^{2}+6+11w=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$3w^{2}+11w+6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3w^{2}+aw+bw+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=3\times 6=18$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 18.

$$1,18 2,9 3,6$$

Tātaihia te tapeke mō ia takirua.

$$1+18=19 2+9=11 3+6=9$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=2 b=9$$

Tuhia anō te 3w^{2}+11w+6 hei \left(3w^{2}+2w\right)+\left(9w+6\right).

$$\left(3w^{2}+2w\right)+\left(9w+6\right)$$

Tauwehea te w i te tuatahi me te 3 i te rōpū tuarua.

$$w\left(3w+2\right)+3\left(3w+2\right)$$

Whakatauwehea atu te kīanga pātahi 3w+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3w+2\right)\left(w+3\right)$$

Hei kimi otinga whārite, me whakaoti te 3w+2=0 me te w+3=0.

$$w=-\frac{2}{3} w=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 11w ki ngā taha e rua.

$$3w^{2}+6+11w=0$$

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.

$$3w^{2}+11w+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 11 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-11±\sqrt{11^{2}-4\times 3\times 6}}{2\times 3}$$

Pūrua 11.

$$w=\frac{-11±\sqrt{121-4\times 3\times 6}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$w=\frac{-11±\sqrt{121-12\times 6}}{2\times 3}$$

Whakareatia -12 ki te 6.

$$w=\frac{-11±\sqrt{121-72}}{2\times 3}$$

Tāpiri 121 ki te -72.

$$w=\frac{-11±\sqrt{49}}{2\times 3}$$

Tuhia te pūtakerua o te 49.

$$w=\frac{-11±7}{2\times 3}$$

Whakareatia 2 ki te 3.

$$w=\frac{-11±7}{6}$$

Nā, me whakaoti te whārite w=\frac{-11±7}{6} ina he tāpiri te ±. Tāpiri -11 ki te 7.

$$w=-\frac{4}{6}$$

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

$$w=-\frac{2}{3}$$

Nā, me whakaoti te whārite w=\frac{-11±7}{6} ina he tango te ±. Tango 7 mai i -11.

$$w=-\frac{18}{6}$$

Whakawehe -18 ki te 6.

$$w=-3$$

Kua oti te whārite te whakatau.

$$w=-\frac{2}{3} w=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 11w ki ngā taha e rua.

$$3w^{2}+6+11w=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3w^{2}+11w=-6$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3w^{2}+11w}{3}=-\frac{6}{3}$$

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

$$w^{2}+\frac{11}{3}w=-\frac{6}{3}$$

Whakawehe -6 ki te 3.

$$w^{2}+\frac{11}{3}w=-2$$

Whakawehea te \frac{11}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{11}{6}. Nā, tāpiria te pūrua o te \frac{11}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$w^{2}+\frac{11}{3}w+\left(\frac{11}{6}\right)^{2}=-2+\left(\frac{11}{6}\right)^{2}$$

Pūruatia \frac{11}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$w^{2}+\frac{11}{3}w+\frac{121}{36}=-2+\frac{121}{36}$$

Tāpiri -2 ki te \frac{121}{36}.

$$w^{2}+\frac{11}{3}w+\frac{121}{36}=\frac{49}{36}$$

Tauwehea w^{2}+\frac{11}{3}w+\frac{121}{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}.

$$\left(w+\frac{11}{6}\right)^{2}=\frac{49}{36}$$

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

$$\sqrt{\left(w+\frac{11}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Whakarūnātia.

$$w+\frac{11}{6}=\frac{7}{6} w+\frac{11}{6}=-\frac{7}{6}$$

Me tango \frac{11}{6} mai i ngā taha e rua o te whārite.

$$w=-\frac{2}{3} w=-3$$