Whakaoti i te 11y=3y^2-4

$$11 y = 3 y ^ { 2 } - 4$$

Whakaoti mō y

$$y=4$$

$$y=-\frac{1}{3}\approx -0.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3y^{2} mai i ngā taha e rua.

$$11y-3y^{2}=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$11y-3y^{2}+4=0$$

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

$$-3y^{2}+11y+4=0$$

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.

$$a+b=11 ab=-3\times 4=-12$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. 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 11.

$$a=12 b=-1$$

Tuhia anō te -3y^{2}+11y+4 hei \left(-3y^{2}+12y\right)+\left(-y+4\right).

$$\left(-3y^{2}+12y\right)+\left(-y+4\right)$$

Whakatauwehea atu 3y i te -3y^{2}+12y.

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

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

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

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3y^{2} mai i ngā taha e rua.

$$11y-3y^{2}=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$11y-3y^{2}+4=0$$

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.

$$-3y^{2}+11y+4=0$$

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

$$y=\frac{-11±\sqrt{11^{2}-4\left(-3\right)\times 4}}{2\left(-3\right)}$$

Pūrua 11.

$$y=\frac{-11±\sqrt{121-4\left(-3\right)\times 4}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$y=\frac{-11±\sqrt{121+12\times 4}}{2\left(-3\right)}$$

Whakareatia 12 ki te 4.

$$y=\frac{-11±\sqrt{121+48}}{2\left(-3\right)}$$

Tāpiri 121 ki te 48.

$$y=\frac{-11±\sqrt{169}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 169.

$$y=\frac{-11±13}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$y=\frac{-11±13}{-6}$$

Nā, me whakaoti te whārite y=\frac{-11±13}{-6} ina he tāpiri te ±. Tāpiri -11 ki te 13.

$$y=\frac{2}{-6}$$

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

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

Nā, me whakaoti te whārite y=\frac{-11±13}{-6} ina he tango te ±. Tango 13 mai i -11.

$$y=-\frac{24}{-6}$$

Whakawehe -24 ki te -6.

$$y=4$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3y^{2} mai i ngā taha e rua.

$$11y-3y^{2}=-4$$

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.

$$-3y^{2}+11y=-4$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3y^{2}+11y}{-3}=-\frac{4}{-3}$$

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

$$y^{2}+\frac{11}{-3}y=-\frac{4}{-3}$$

Whakawehe 11 ki te -3.

$$y^{2}-\frac{11}{3}y=-\frac{4}{-3}$$

Whakawehe -4 ki te -3.

$$y^{2}-\frac{11}{3}y=\frac{4}{3}$$

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.

$$y^{2}-\frac{11}{3}y+\left(-\frac{11}{6}\right)^{2}=\frac{4}{3}+\left(-\frac{11}{6}\right)^{2}$$

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

$$y^{2}-\frac{11}{3}y+\frac{121}{36}=\frac{4}{3}+\frac{121}{36}$$

Tāpiri \frac{4}{3} ki te \frac{121}{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.

$$y^{2}-\frac{11}{3}y+\frac{121}{36}=\frac{169}{36}$$

Tauwehea y^{2}-\frac{11}{3}y+\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}.

$$\left(y-\frac{11}{6}\right)^{2}=\frac{169}{36}$$

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

$$\sqrt{\left(y-\frac{11}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Whakarūnātia.

$$y-\frac{11}{6}=\frac{13}{6} y-\frac{11}{6}=-\frac{13}{6}$$

Me tāpiri \frac{11}{6} ki ngā taha e rua o te whārite.

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