Whakaoti i te y^2+6y+9

$$y ^ { 2 } + 6 y + 9$$

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei y^{2}+ay+by+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=1\times 9=9$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

$$1,9 3,3$$

Tātaihia te tapeke mō ia takirua.

$$1+9=10 3+3=6$$

Ko te otinga te takirua ka hoatu i te tapeke 6.

$$a=3 b=3$$

Tuhia anō te y^{2}+6y+9 hei \left(y^{2}+3y\right)+\left(3y+9\right).

$$\left(y^{2}+3y\right)+\left(3y+9\right)$$

Tauwehea te y i te tuatahi me te 3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi y+3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Tuhia anōtia hei pūrua huarua.

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

Ngā Upane Whakamahi Tauwehe o te Pūrua Huarua

Ko te tikanga tātai o tēnei huatoru he pūrua huatoru, ka whakareatia pea e tētahi tauwehe pātahi. Ka taea ngā pūrua huatoru te tauwehe mā te kimi i ngā pūtakerua o ngā kīanga tau ārahi, autō hoki.

$$factor(y^{2}+6y+9)$$

Kimihia te pūtakerua o te kīanga tau autō, 9.

$$\sqrt{9}=3$$

Ko te pūrua huatoru te pūrua o te huarua ko te tapeke tērā, te huatango rānei o ngā pūtakerua o ngā kīanga tau ārahi, autō hoki, e whakaritea ai te tohu e te tohu o te kīanga tau waenga o te pūrua huatoru.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$y^{2}+6y+9=0$$

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.

$$y=\frac{-6±\sqrt{6^{2}-4\times 9}}{2}$$

Pūrua 6.

$$y=\frac{-6±\sqrt{36-4\times 9}}{2}$$

Whakareatia -4 ki te 9.

$$y=\frac{-6±\sqrt{36-36}}{2}$$

Tāpiri 36 ki te -36.

$$y=\frac{-6±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

$$y=\frac{-6±0}{2}$$

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te -3 mō te x_{1} me te -3 mō te x_{2}.

$$y^{2}+6y+9=\left(y-\left(-3\right)\right)\left(y-\left(-3\right)\right)$$

Whakamāmātia ngā kīanga katoa o te āhua p-\left(-q\right) ki te p+q.

$$y^{2}+6y+9=\left(y+3\right)\left(y+3\right)$$

Aromātai

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