Whakaoti i te 64v^2+48v+9

$$64 v ^ { 2 } + 48 v + 9$$

Tauwehe

$$\left(8v+3\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 64v^{2}+av+bv+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 tōrunga te a+b, he tōrunga 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 64v^{2}+48v+9 hei \left(64v^{2}+24v\right)+\left(24v+9\right).

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

Tauwehea te 8v i te tuatahi me te 3 i te rōpū tuarua.

$$8v\left(8v+3\right)+3\left(8v+3\right)$$

Whakatauwehea atu te kīanga pātahi 8v+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(8v+3\right)\left(8v+3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(8v+3\right)^{2}$$

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.

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

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

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

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

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

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.

$$\left(8v+3\right)^{2}$$

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.

$$64v^{2}+48v+9=0$$

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.

$$v=\frac{-48±\sqrt{48^{2}-4\times 64\times 9}}{2\times 64}$$

Pūrua 48.

$$v=\frac{-48±\sqrt{2304-4\times 64\times 9}}{2\times 64}$$

Whakareatia -4 ki te 64.

$$v=\frac{-48±\sqrt{2304-256\times 9}}{2\times 64}$$

Whakareatia -256 ki te 9.

$$v=\frac{-48±\sqrt{2304-2304}}{2\times 64}$$

Tāpiri 2304 ki te -2304.

$$v=\frac{-48±\sqrt{0}}{2\times 64}$$

Tuhia te pūtakerua o te 0.

$$v=\frac{-48±0}{2\times 64}$$

Whakareatia 2 ki te 64.

$$v=\frac{-48±0}{128}$$

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}.

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

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

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

Tāpiri \frac{3}{8} ki te v 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.

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

Tāpiri \frac{3}{8} ki te v 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.

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

Whakareatia \frac{8v+3}{8} ki te \frac{8v+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.

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

Whakareatia 8 ki te 8.

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

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

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

Aromātai

$$\left(8v+3\right)^{2}$$