Whakaoti i te -y^2+10=3y

$$- y ^ { 2 } + 10 = 3 y$$

Whakaoti mō y

$$y=-5$$

$$y=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3y mai i ngā taha e rua.

$$-y^{2}+10-3y=0$$

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

$$-y^{2}-3y+10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -y^{2}+ay+by+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=-10=-10$$

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 -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=2 b=-5$$

Tuhia anō te -y^{2}-3y+10 hei \left(-y^{2}+2y\right)+\left(-5y+10\right).

$$\left(-y^{2}+2y\right)+\left(-5y+10\right)$$

Tauwehea te y i te tuatahi me te 5 i te rōpū tuarua.

$$y\left(-y+2\right)+5\left(-y+2\right)$$

Whakatauwehea atu te kīanga pātahi -y+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-y+2\right)\left(y+5\right)$$

Hei kimi otinga whārite, me whakaoti te -y+2=0 me te y+5=0.

$$y=2 y=-5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3y mai i ngā taha e rua.

$$-y^{2}+10-3y=0$$

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.

$$-y^{2}-3y+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -3 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Pūrua -3.

$$y=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$y=\frac{-\left(-3\right)±\sqrt{9+4\times 10}}{2\left(-1\right)}$$

Whakareatia 4 ki te 10.

$$y=\frac{-\left(-3\right)±\sqrt{9+40}}{2\left(-1\right)}$$

Tāpiri 9 ki te 40.

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

Tuhia te pūtakerua o te 49.

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

Ko te tauaro o -3 ko 3.

$$y=\frac{3±7}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$y=\frac{3±7}{-2}$$

Nā, me whakaoti te whārite y=\frac{3±7}{-2} ina he tāpiri te ±. Tāpiri 3 ki te 7.

$$y=\frac{10}{-2}$$

Whakawehe 10 ki te -2.

$$y=-5$$

Nā, me whakaoti te whārite y=\frac{3±7}{-2} ina he tango te ±. Tango 7 mai i 3.

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

Whakawehe -4 ki te -2.

$$y=2$$

Kua oti te whārite te whakatau.

$$y=-5 y=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3y mai i ngā taha e rua.

$$-y^{2}+10-3y=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-y^{2}-3y=-10$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-y^{2}-3y}{-1}=-\frac{10}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$y^{2}+\left(-\frac{3}{-1}\right)y=-\frac{10}{-1}$$

Whakawehe -3 ki te -1.

$$y^{2}+3y=-\frac{10}{-1}$$

Whakawehe -10 ki te -1.

$$y^{2}+3y=10$$

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

$$y^{2}+3y+\left(\frac{3}{2}\right)^{2}=10+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}+3y+\frac{9}{4}=10+\frac{9}{4}$$

Tāpiri 10 ki te \frac{9}{4}.

$$y^{2}+3y+\frac{9}{4}=\frac{49}{4}$$

Tauwehea y^{2}+3y+\frac{9}{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}.

$$\left(y+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(y+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$y+\frac{3}{2}=\frac{7}{2} y+\frac{3}{2}=-\frac{7}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$y=2 y=-5$$