Whakaoti i te y+2y^2+y=4

$$y+2 { y }^{ 2 } +y = 4$$

Whakaoti mō y

$$y=-2$$

$$y=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Pahekotia te y me y, ka 2y.

$$2y+2y^{2}=4$$

Tangohia te 4 mai i ngā taha e rua.

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

Whakawehea ngā taha e rua ki te 2.

$$y+y^{2}-2=0$$

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

$$y^{2}+y-2=0$$

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

$$a+b=1 ab=1\left(-2\right)=-2$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-1 b=2$$

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

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

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

$$y\left(y-1\right)+2\left(y-1\right)$$

Whakatauwehea atu te kīanga pātahi y-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-1\right)\left(y+2\right)$$

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

$$y=1 y=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Pahekotia te y me y, ka 2y.

$$2y+2y^{2}=4$$

Tangohia te 4 mai i ngā taha e rua.

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

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.

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

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

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

Pūrua 2.

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

Whakareatia -4 ki te 2.

$$y=\frac{-2±\sqrt{4-8\left(-4\right)}}{2\times 2}$$

Whakareatia -8 ki te -4.

$$y=\frac{-2±\sqrt{4+32}}{2\times 2}$$

Tāpiri 4 ki te 32.

$$y=\frac{-2±\sqrt{36}}{2\times 2}$$

Tuhia te pūtakerua o te 36.

$$y=\frac{-2±6}{2\times 2}$$

Whakareatia 2 ki te 2.

$$y=\frac{-2±6}{4}$$

Nā, me whakaoti te whārite y=\frac{-2±6}{4} ina he tāpiri te ±. Tāpiri -2 ki te 6.

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

Whakawehe 4 ki te 4.

$$y=1$$

Nā, me whakaoti te whārite y=\frac{-2±6}{4} ina he tango te ±. Tango 6 mai i -2.

$$y=-\frac{8}{4}$$

Whakawehe -8 ki te 4.

$$y=-2$$

Kua oti te whārite te whakatau.

$$y=1 y=-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Pahekotia te y me y, ka 2y.

$$2y+2y^{2}=4$$

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.

$$2y^{2}+2y=4$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2y^{2}+2y}{2}=\frac{4}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$y^{2}+\frac{2}{2}y=\frac{4}{2}$$

Whakawehe 2 ki te 2.

$$y^{2}+y=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$y^{2}+y=2$$

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

$$y^{2}+y+\left(\frac{1}{2}\right)^{2}=2+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

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

Tāpiri 2 ki te \frac{1}{4}.

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

Tauwehea y^{2}+y+\frac{1}{4}. 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}.

$$\left(y+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(y+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

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

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$y=1 y=-2$$