Whakaoti i te 36+121c^2-132c=

$$36 + 121 c ^ { 2 } - 132 c =$$

Tauwehe

$$\left(11c-6\right)^{2}$$

Ngā Upane Otinga

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

$$121c^{2}-132c+36$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 121c^{2}+ac+bc+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-132 ab=121\times 36=4356$$

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 4356.

$$-1,-4356 -2,-2178 -3,-1452 -4,-1089 -6,-726 -9,-484 -11,-396 -12,-363 -18,-242 -22,-198 -33,-132 -36,-121 -44,-99 -66,-66$$

Tātaihia te tapeke mō ia takirua.

$$-1-4356=-4357 -2-2178=-2180 -3-1452=-1455 -4-1089=-1093 -6-726=-732 -9-484=-493 -11-396=-407 -12-363=-375 -18-242=-260 -22-198=-220 -33-132=-165 -36-121=-157 -44-99=-143 -66-66=-132$$

Ko te otinga te takirua ka hoatu i te tapeke -132.

$$a=-66 b=-66$$

Tuhia anō te 121c^{2}-132c+36 hei \left(121c^{2}-66c\right)+\left(-66c+36\right).

$$\left(121c^{2}-66c\right)+\left(-66c+36\right)$$

Tauwehea te 11c i te tuatahi me te -6 i te rōpū tuarua.

$$11c\left(11c-6\right)-6\left(11c-6\right)$$

Whakatauwehea atu te kīanga pātahi 11c-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(11c-6\right)\left(11c-6\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(11c-6\right)^{2}$$

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.

$$factor(121c^{2}-132c+36)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(121,-132,36)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 121c^{2}.

$$\sqrt{121c^{2}}=11c$$

Kimihia te pūtakerua o te kīanga tau autō, 36.

$$\sqrt{36}=6$$

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.

$$\left(11c-6\right)^{2}$$

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.

$$121c^{2}-132c+36=0$$

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.

$$c=\frac{-\left(-132\right)±\sqrt{\left(-132\right)^{2}-4\times 121\times 36}}{2\times 121}$$

Pūrua -132.

$$c=\frac{-\left(-132\right)±\sqrt{17424-4\times 121\times 36}}{2\times 121}$$

Whakareatia -4 ki te 121.

$$c=\frac{-\left(-132\right)±\sqrt{17424-484\times 36}}{2\times 121}$$

Whakareatia -484 ki te 36.

$$c=\frac{-\left(-132\right)±\sqrt{17424-17424}}{2\times 121}$$

Tāpiri 17424 ki te -17424.

$$c=\frac{-\left(-132\right)±\sqrt{0}}{2\times 121}$$

Tuhia te pūtakerua o te 0.

$$c=\frac{-\left(-132\right)±0}{2\times 121}$$

Ko te tauaro o -132 ko 132.

$$c=\frac{132±0}{2\times 121}$$

Whakareatia 2 ki te 121.

$$c=\frac{132±0}{242}$$

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{6}{11} mō te x_{1} me te \frac{6}{11} mō te x_{2}.

$$121c^{2}-132c+36=121\left(c-\frac{6}{11}\right)\left(c-\frac{6}{11}\right)$$

Tango \frac{6}{11} mai i c 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.

$$121c^{2}-132c+36=121\times \frac{11c-6}{11}\left(c-\frac{6}{11}\right)$$

Tango \frac{6}{11} mai i c 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.

$$121c^{2}-132c+36=121\times \frac{11c-6}{11}\times \frac{11c-6}{11}$$

Whakareatia \frac{11c-6}{11} ki te \frac{11c-6}{11} 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.

$$121c^{2}-132c+36=121\times \frac{\left(11c-6\right)\left(11c-6\right)}{11\times 11}$$

Whakareatia 11 ki te 11.

$$121c^{2}-132c+36=121\times \frac{\left(11c-6\right)\left(11c-6\right)}{121}$$

Whakakorea atu te tauwehe pūnoa nui rawa 121 i roto i te 121 me te 121.

$$121c^{2}-132c+36=\left(11c-6\right)\left(11c-6\right)$$

Aromātai

$$\left(11c-6\right)^{2}$$