Whakaoti i te 3y^2-y-4=0

$$3 y ^ { 2 } - y - 4 = 0$$

Whakaoti mō y

$$y=-1$$

$$y = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3y^{2}+ay+by-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=3\left(-4\right)=-12$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-4 b=3$$

Tuhia anō te 3y^{2}-y-4 hei \left(3y^{2}-4y\right)+\left(3y-4\right).

$$\left(3y^{2}-4y\right)+\left(3y-4\right)$$

Whakatauwehea atu y i te 3y^{2}-4y.

$$y\left(3y-4\right)+3y-4$$

Whakatauwehea atu te kīanga pātahi 3y-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3y-4\right)\left(y+1\right)$$

Hei kimi otinga whārite, me whakaoti te 3y-4=0 me te y+1=0.

$$y=\frac{4}{3} y=-1$$

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.

$$3y^{2}-y-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -1 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-4\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$y=\frac{-\left(-1\right)±\sqrt{1-12\left(-4\right)}}{2\times 3}$$

Whakareatia -12 ki te -4.

$$y=\frac{-\left(-1\right)±\sqrt{1+48}}{2\times 3}$$

Tāpiri 1 ki te 48.

$$y=\frac{-\left(-1\right)±\sqrt{49}}{2\times 3}$$

Tuhia te pūtakerua o te 49.

$$y=\frac{-\left(-1\right)±7}{2\times 3}$$

Ko te tauaro o -1 ko 1.

$$y=\frac{1±7}{2\times 3}$$

Whakareatia 2 ki te 3.

$$y=\frac{1±7}{6}$$

Nā, me whakaoti te whārite y=\frac{1±7}{6} ina he tāpiri te ±. Tāpiri 1 ki te 7.

$$y=\frac{8}{6}$$

Whakahekea te hautanga \frac{8}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=\frac{4}{3}$$

Nā, me whakaoti te whārite y=\frac{1±7}{6} ina he tango te ±. Tango 7 mai i 1.

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

Whakawehe -6 ki te 6.

$$y=-1$$

Kua oti te whārite te whakatau.

$$y=\frac{4}{3} y=-1$$

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.

$$3y^{2}-y-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$3y^{2}-y-4-\left(-4\right)=-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$3y^{2}-y=-\left(-4\right)$$

Tango -4 mai i 0.

$$3y^{2}-y=4$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3y^{2}-y}{3}=\frac{4}{3}$$

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

$$y^{2}-\frac{1}{3}y=\frac{4}{3}$$

Whakawehea te -\frac{1}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{6}. Nā, tāpiria te pūrua o te -\frac{1}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{1}{3}y+\left(-\frac{1}{6}\right)^{2}=\frac{4}{3}+\left(-\frac{1}{6}\right)^{2}$$

Pūruatia -\frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{1}{3}y+\frac{1}{36}=\frac{4}{3}+\frac{1}{36}$$

Tāpiri \frac{4}{3} ki te \frac{1}{36} 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.

$$y^{2}-\frac{1}{3}y+\frac{1}{36}=\frac{49}{36}$$

Tauwehea y^{2}-\frac{1}{3}y+\frac{1}{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}.

$$\left(y-\frac{1}{6}\right)^{2}=\frac{49}{36}$$

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

$$\sqrt{\left(y-\frac{1}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Whakarūnātia.

$$y-\frac{1}{6}=\frac{7}{6} y-\frac{1}{6}=-\frac{7}{6}$$

Me tāpiri \frac{1}{6} ki ngā taha e rua o te whārite.

$$y=\frac{4}{3} y=-1$$