Whakaoti i te 70y+49+25y^2=

$$70 y + 49 + 25 y ^ { 2 } =$$

Tauwehe

$$\left(5y+7\right)^{2}$$

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.

$$25y^{2}+70y+49$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 25y^{2}+ay+by+49. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=70 ab=25\times 49=1225$$

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 1225.

$$1,1225 5,245 7,175 25,49 35,35$$

Tātaihia te tapeke mō ia takirua.

$$1+1225=1226 5+245=250 7+175=182 25+49=74 35+35=70$$

Ko te otinga te takirua ka hoatu i te tapeke 70.

$$a=35 b=35$$

Tuhia anō te 25y^{2}+70y+49 hei \left(25y^{2}+35y\right)+\left(35y+49\right).

$$\left(25y^{2}+35y\right)+\left(35y+49\right)$$

Tauwehea te 5y i te tuatahi me te 7 i te rōpū tuarua.

$$5y\left(5y+7\right)+7\left(5y+7\right)$$

Whakatauwehea atu te kīanga pātahi 5y+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5y+7\right)\left(5y+7\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(5y+7\right)^{2}$$

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.

$$factor(25y^{2}+70y+49)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(25,70,49)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 25y^{2}.

$$\sqrt{25y^{2}}=5y$$

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.

$$\left(5y+7\right)^{2}$$

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.

$$25y^{2}+70y+49=0$$

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.

$$y=\frac{-70±\sqrt{70^{2}-4\times 25\times 49}}{2\times 25}$$

Pūrua 70.

$$y=\frac{-70±\sqrt{4900-4\times 25\times 49}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$y=\frac{-70±\sqrt{4900-100\times 49}}{2\times 25}$$

Whakareatia -100 ki te 49.

$$y=\frac{-70±\sqrt{4900-4900}}{2\times 25}$$

Tāpiri 4900 ki te -4900.

$$y=\frac{-70±\sqrt{0}}{2\times 25}$$

Tuhia te pūtakerua o te 0.

$$y=\frac{-70±0}{2\times 25}$$

Whakareatia 2 ki te 25.

$$y=\frac{-70±0}{50}$$

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}{5} mō te x_{1} me te -\frac{7}{5} mō te x_{2}.

$$25y^{2}+70y+49=25\left(y-\left(-\frac{7}{5}\right)\right)\left(y-\left(-\frac{7}{5}\right)\right)$$

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

$$25y^{2}+70y+49=25\left(y+\frac{7}{5}\right)\left(y+\frac{7}{5}\right)$$

Tāpiri \frac{7}{5} ki te y mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$25y^{2}+70y+49=25\times \frac{5y+7}{5}\left(y+\frac{7}{5}\right)$$

Tāpiri \frac{7}{5} ki te y mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$25y^{2}+70y+49=25\times \frac{5y+7}{5}\times \frac{5y+7}{5}$$

Whakareatia \frac{5y+7}{5} ki te \frac{5y+7}{5} 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.

$$25y^{2}+70y+49=25\times \frac{\left(5y+7\right)\left(5y+7\right)}{5\times 5}$$

Whakareatia 5 ki te 5.

$$25y^{2}+70y+49=25\times \frac{\left(5y+7\right)\left(5y+7\right)}{25}$$

Whakakorea atu te tauwehe pūnoa nui rawa 25 i roto i te 25 me te 25.

$$25y^{2}+70y+49=\left(5y+7\right)\left(5y+7\right)$$

Aromātai

$$\left(5y+7\right)^{2}$$