Whakaoti i te 64x^2-48x+9=

$$64 x ^ { 2 } - 48 x + 9 =$$

Tauwehe

$$\left(8x-3\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 64x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-48 ab=64\times 9=576$$

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 576.

$$-1,-576 -2,-288 -3,-192 -4,-144 -6,-96 -8,-72 -9,-64 -12,-48 -16,-36 -18,-32 -24,-24$$

Tātaihia te tapeke mō ia takirua.

$$-1-576=-577 -2-288=-290 -3-192=-195 -4-144=-148 -6-96=-102 -8-72=-80 -9-64=-73 -12-48=-60 -16-36=-52 -18-32=-50 -24-24=-48$$

Ko te otinga te takirua ka hoatu i te tapeke -48.

$$a=-24 b=-24$$

Tuhia anō te 64x^{2}-48x+9 hei \left(64x^{2}-24x\right)+\left(-24x+9\right).

$$\left(64x^{2}-24x\right)+\left(-24x+9\right)$$

Tauwehea te 8x i te tuatahi me te -3 i te rōpū tuarua.

$$8x\left(8x-3\right)-3\left(8x-3\right)$$

Whakatauwehea atu te kīanga pātahi 8x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(8x-3\right)\left(8x-3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(8x-3\right)^{2}$$

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.

$$factor(64x^{2}-48x+9)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(64,-48,9)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 64x^{2}.

$$\sqrt{64x^{2}}=8x$$

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.

$$\left(8x-3\right)^{2}$$

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.

$$64x^{2}-48x+9=0$$

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.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 64\times 9}}{2\times 64}$$

Pūrua -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 64\times 9}}{2\times 64}$$

Whakareatia -4 ki te 64.

$$x=\frac{-\left(-48\right)±\sqrt{2304-256\times 9}}{2\times 64}$$

Whakareatia -256 ki te 9.

$$x=\frac{-\left(-48\right)±\sqrt{2304-2304}}{2\times 64}$$

Tāpiri 2304 ki te -2304.

$$x=\frac{-\left(-48\right)±\sqrt{0}}{2\times 64}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-\left(-48\right)±0}{2\times 64}$$

Ko te tauaro o -48 ko 48.

$$x=\frac{48±0}{2\times 64}$$

Whakareatia 2 ki te 64.

$$x=\frac{48±0}{128}$$

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}{8} mō te x_{1} me te \frac{3}{8} mō te x_{2}.

$$64x^{2}-48x+9=64\left(x-\frac{3}{8}\right)\left(x-\frac{3}{8}\right)$$

Tango \frac{3}{8} 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.

$$64x^{2}-48x+9=64\times \frac{8x-3}{8}\left(x-\frac{3}{8}\right)$$

Tango \frac{3}{8} 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.

$$64x^{2}-48x+9=64\times \frac{8x-3}{8}\times \frac{8x-3}{8}$$

Whakareatia \frac{8x-3}{8} ki te \frac{8x-3}{8} 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.

$$64x^{2}-48x+9=64\times \frac{\left(8x-3\right)\left(8x-3\right)}{8\times 8}$$

Whakareatia 8 ki te 8.

$$64x^{2}-48x+9=64\times \frac{\left(8x-3\right)\left(8x-3\right)}{64}$$

Whakakorea atu te tauwehe pūnoa nui rawa 64 i roto i te 64 me te 64.

$$64x^{2}-48x+9=\left(8x-3\right)\left(8x-3\right)$$

Aromātai

$$\left(8x-3\right)^{2}$$