Whakaoti i te 49x^2-42x+9

$$49 x ^ { 2 } - 42 x + 9$$

Tauwehe

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

Ngā Upane Otinga

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

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

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 49x^{2}-42x+9 hei \left(49x^{2}-21x\right)+\left(-21x+9\right).

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

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

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

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

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

Tuhia anōtia hei pūrua huarua.

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

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.

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

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

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

Kimihia te pūtakerua o te kīanga tau ārahi, 49x^{2}.

$$\sqrt{49x^{2}}=7x$$

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.

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

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.

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

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.

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

Pūrua -42.

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

Whakareatia -4 ki te 49.

$$x=\frac{-\left(-42\right)±\sqrt{1764-196\times 9}}{2\times 49}$$

Whakareatia -196 ki te 9.

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

Tāpiri 1764 ki te -1764.

$$x=\frac{-\left(-42\right)±\sqrt{0}}{2\times 49}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-\left(-42\right)±0}{2\times 49}$$

Ko te tauaro o -42 ko 42.

$$x=\frac{42±0}{2\times 49}$$

Whakareatia 2 ki te 49.

$$x=\frac{42±0}{98}$$

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

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

Tango \frac{3}{7} mai i x 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.

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

Tango \frac{3}{7} mai i x 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.

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

Whakareatia \frac{7x-3}{7} ki te \frac{7x-3}{7} 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.

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

Whakareatia 7 ki te 7.

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

Whakakorea atu te tauwehe pūnoa nui rawa 49 i roto i te 49 me te 49.

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

Aromātai

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