Whakaoti i te y^2-8y+12=0

$$y ^ { 2 } - 8 y + 12 = 0$$

Whakaoti mō y

$$y=2$$

$$y=6$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te y^{2}-8y+12 mā te whakamahi i te tātai y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=12$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. 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.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-6 b=-2$$

Me tuhi anō te kīanga whakatauwehe \left(y+a\right)\left(y+b\right) mā ngā uara i tātaihia.

$$\left(y-6\right)\left(y-2\right)$$

Hei kimi otinga whārite, me whakaoti te y-6=0 me te y-2=0.

$$y=6 y=2$$

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 y^{2}+ay+by+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\times 12=12$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. 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.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-6 b=-2$$

Tuhia anō te y^{2}-8y+12 hei \left(y^{2}-6y\right)+\left(-2y+12\right).

$$\left(y^{2}-6y\right)+\left(-2y+12\right)$$

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

$$y\left(y-6\right)-2\left(y-6\right)$$

Whakatauwehea atu te kīanga pātahi y-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-6\right)\left(y-2\right)$$

Hei kimi otinga whārite, me whakaoti te y-6=0 me te y-2=0.

$$y=6 y=2$$

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.

$$y^{2}-8y+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 12}}{2}$$

Pūrua -8.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\times 12}}{2}$$

Whakareatia -4 ki te 12.

$$y=\frac{-\left(-8\right)±\sqrt{64-48}}{2}$$

Tāpiri 64 ki te -48.

$$y=\frac{-\left(-8\right)±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$y=\frac{-\left(-8\right)±4}{2}$$

Ko te tauaro o -8 ko 8.

$$y=\frac{8±4}{2}$$

Nā, me whakaoti te whārite y=\frac{8±4}{2} ina he tāpiri te ±. Tāpiri 8 ki te 4.

$$y=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$y=6$$

Nā, me whakaoti te whārite y=\frac{8±4}{2} ina he tango te ±. Tango 4 mai i 8.

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

Whakawehe 4 ki te 2.

$$y=2$$

Kua oti te whārite te whakatau.

$$y=6 y=2$$

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.

$$y^{2}-8y+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$y^{2}-8y+12-12=-12$$

Mā te tango i te 12 i a ia ake anō ka toe ko te 0.

$$y^{2}-8y=-12$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-8y+\left(-4\right)^{2}=-12+\left(-4\right)^{2}$$

Pūrua -4.

$$y^{2}-8y+16=-12+16$$

Tāpiri -12 ki te 16.

$$y^{2}-8y+16=4$$

Tauwehea y^{2}-8y+16. 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}.

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

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

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

Whakarūnātia.

$$y-4=2 y-4=-2$$

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

$$y=6 y=2$$