Whakaoti i te (3+2y)^2+2y^2=3

$$( 3 + 2 y ) ^ { 2 } + 2 y ^ { 2 } = 3$$

Whakaoti mō y

$$y=-1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3+2y\right)^{2}.

$$9+12y+4y^{2}+2y^{2}=3$$

Pahekotia te 4y^{2} me 2y^{2}, ka 6y^{2}.

$$9+12y+6y^{2}=3$$

Tangohia te 3 mai i ngā taha e rua.

$$9+12y+6y^{2}-3=0$$

Tangohia te 3 i te 9, ka 6.

$$6+12y+6y^{2}=0$$

Whakawehea ngā taha e rua ki te 6.

$$1+2y+y^{2}=0$$

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

$$y^{2}+2y+1=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei y^{2}+ay+by+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

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

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.

$$a=1 b=1$$

Tuhia anō te y^{2}+2y+1 hei \left(y^{2}+y\right)+\left(y+1\right).

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

Whakatauwehea atu y i te y^{2}+y.

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

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

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

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te y+1=0.

$$y=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3+2y\right)^{2}.

$$9+12y+4y^{2}+2y^{2}=3$$

Pahekotia te 4y^{2} me 2y^{2}, ka 6y^{2}.

$$9+12y+6y^{2}=3$$

Tangohia te 3 mai i ngā taha e rua.

$$9+12y+6y^{2}-3=0$$

Tangohia te 3 i te 9, ka 6.

$$6+12y+6y^{2}=0$$

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.

$$6y^{2}+12y+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 12 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-12±\sqrt{12^{2}-4\times 6\times 6}}{2\times 6}$$

Pūrua 12.

$$y=\frac{-12±\sqrt{144-4\times 6\times 6}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$y=\frac{-12±\sqrt{144-24\times 6}}{2\times 6}$$

Whakareatia -24 ki te 6.

$$y=\frac{-12±\sqrt{144-144}}{2\times 6}$$

Tāpiri 144 ki te -144.

$$y=\frac{-12±\sqrt{0}}{2\times 6}$$

Tuhia te pūtakerua o te 0.

$$y=-\frac{12}{2\times 6}$$

Whakareatia 2 ki te 6.

$$y=-\frac{12}{12}$$

Whakawehe -12 ki te 12.

$$y=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3+2y\right)^{2}.

$$9+12y+4y^{2}+2y^{2}=3$$

Pahekotia te 4y^{2} me 2y^{2}, ka 6y^{2}.

$$9+12y+6y^{2}=3$$

Tangohia te 9 mai i ngā taha e rua.

$$12y+6y^{2}=3-9$$

Tangohia te 9 i te 3, ka -6.

$$12y+6y^{2}=-6$$

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.

$$6y^{2}+12y=-6$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6y^{2}+12y}{6}=-\frac{6}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$y^{2}+\frac{12}{6}y=-\frac{6}{6}$$

Whakawehe 12 ki te 6.

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

Whakawehe -6 ki te 6.

$$y^{2}+2y=-1$$

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

$$y^{2}+2y+1^{2}=-1+1^{2}$$

Pūrua 1.

$$y^{2}+2y+1=-1+1$$

Tāpiri -1 ki te 1.

$$y^{2}+2y+1=0$$

Tauwehea y^{2}+2y+1. 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}.

$$\left(y+1\right)^{2}=0$$

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

$$\sqrt{\left(y+1\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$y+1=0 y+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$y=-1 y=-1$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$y=-1$$