Whakaoti i te 9z^2-42z+49=

$$9 z ^ { 2 } - 42 z + 49 =$$

Tauwehe

$$\left(3z-7\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9z^{2}+az+bz+49. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-42 ab=9\times 49=441$$

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 441.

$$-1,-441 -3,-147 -7,-63 -9,-49 -21,-21$$

Tātaihia te tapeke mō ia takirua.

$$-1-441=-442 -3-147=-150 -7-63=-70 -9-49=-58 -21-21=-42$$

Ko te otinga te takirua ka hoatu i te tapeke -42.

$$a=-21 b=-21$$

Tuhia anō te 9z^{2}-42z+49 hei \left(9z^{2}-21z\right)+\left(-21z+49\right).

$$\left(9z^{2}-21z\right)+\left(-21z+49\right)$$

Tauwehea te 3z i te tuatahi me te -7 i te rōpū tuarua.

$$3z\left(3z-7\right)-7\left(3z-7\right)$$

Whakatauwehea atu te kīanga pātahi 3z-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3z-7\right)\left(3z-7\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(3z-7\right)^{2}$$

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.

$$factor(9z^{2}-42z+49)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(9,-42,49)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 9z^{2}.

$$\sqrt{9z^{2}}=3z$$

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.

$$\left(3z-7\right)^{2}$$

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.

$$9z^{2}-42z+49=0$$

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.

$$z=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 9\times 49}}{2\times 9}$$

Pūrua -42.

$$z=\frac{-\left(-42\right)±\sqrt{1764-4\times 9\times 49}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$z=\frac{-\left(-42\right)±\sqrt{1764-36\times 49}}{2\times 9}$$

Whakareatia -36 ki te 49.

$$z=\frac{-\left(-42\right)±\sqrt{1764-1764}}{2\times 9}$$

Tāpiri 1764 ki te -1764.

$$z=\frac{-\left(-42\right)±\sqrt{0}}{2\times 9}$$

Tuhia te pūtakerua o te 0.

$$z=\frac{-\left(-42\right)±0}{2\times 9}$$

Ko te tauaro o -42 ko 42.

$$z=\frac{42±0}{2\times 9}$$

Whakareatia 2 ki te 9.

$$z=\frac{42±0}{18}$$

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 \frac{7}{3} mō te x_{1} me te \frac{7}{3} mō te x_{2}.

$$9z^{2}-42z+49=9\left(z-\frac{7}{3}\right)\left(z-\frac{7}{3}\right)$$

Tango \frac{7}{3} mai i z mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$9z^{2}-42z+49=9\times \frac{3z-7}{3}\left(z-\frac{7}{3}\right)$$

Tango \frac{7}{3} mai i z mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$9z^{2}-42z+49=9\times \frac{3z-7}{3}\times \frac{3z-7}{3}$$

Whakareatia \frac{3z-7}{3} ki te \frac{3z-7}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$9z^{2}-42z+49=9\times \frac{\left(3z-7\right)\left(3z-7\right)}{3\times 3}$$

Whakareatia 3 ki te 3.

$$9z^{2}-42z+49=9\times \frac{\left(3z-7\right)\left(3z-7\right)}{9}$$

Whakakorea atu te tauwehe pūnoa nui rawa 9 i roto i te 9 me te 9.

$$9z^{2}-42z+49=\left(3z-7\right)\left(3z-7\right)$$

Aromātai

$$\left(3z-7\right)^{2}$$