Whakaoti i te p^2+14p+49

$$p ^ { 2 } + 14 p + 49$$

Tauwehe

$$\left(p+7\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei p^{2}+ap+bp+49. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=1\times 49=49$$

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 49.

$$1,49 7,7$$

Tātaihia te tapeke mō ia takirua.

$$1+49=50 7+7=14$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=7 b=7$$

Tuhia anō te p^{2}+14p+49 hei \left(p^{2}+7p\right)+\left(7p+49\right).

$$\left(p^{2}+7p\right)+\left(7p+49\right)$$

Tauwehea te p i te tuatahi me te 7 i te rōpū tuarua.

$$p\left(p+7\right)+7\left(p+7\right)$$

Whakatauwehea atu te kīanga pātahi p+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(p+7\right)\left(p+7\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(p+7\right)^{2}$$

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.

$$factor(p^{2}+14p+49)$$

Kimihia te pūtakerua o te kīanga tau autō, 49.

$$\sqrt{49}=7$$

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.

$$\left(p+7\right)^{2}$$

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.

$$p^{2}+14p+49=0$$

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.

$$p=\frac{-14±\sqrt{14^{2}-4\times 49}}{2}$$

Pūrua 14.

$$p=\frac{-14±\sqrt{196-4\times 49}}{2}$$

Whakareatia -4 ki te 49.

$$p=\frac{-14±\sqrt{196-196}}{2}$$

Tāpiri 196 ki te -196.

$$p=\frac{-14±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

$$p=\frac{-14±0}{2}$$

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 -7 mō te x_{1} me te -7 mō te x_{2}.

$$p^{2}+14p+49=\left(p-\left(-7\right)\right)\left(p-\left(-7\right)\right)$$

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

$$p^{2}+14p+49=\left(p+7\right)\left(p+7\right)$$

Aromātai

$$\left(p+7\right)^{2}$$