Whakaoti i te 49v^2+112v+64

$$49 v ^ { 2 } + 112 v + 64$$

Tauwehe

$$\left(7v+8\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 49v^{2}+av+bv+64. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=112 ab=49\times 64=3136$$

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 3136.

$$1,3136 2,1568 4,784 7,448 8,392 14,224 16,196 28,112 32,98 49,64 56,56$$

Tātaihia te tapeke mō ia takirua.

$$1+3136=3137 2+1568=1570 4+784=788 7+448=455 8+392=400 14+224=238 16+196=212 28+112=140 32+98=130 49+64=113 56+56=112$$

Ko te otinga te takirua ka hoatu i te tapeke 112.

$$a=56 b=56$$

Tuhia anō te 49v^{2}+112v+64 hei \left(49v^{2}+56v\right)+\left(56v+64\right).

$$\left(49v^{2}+56v\right)+\left(56v+64\right)$$

Tauwehea te 7v i te tuatahi me te 8 i te rōpū tuarua.

$$7v\left(7v+8\right)+8\left(7v+8\right)$$

Whakatauwehea atu te kīanga pātahi 7v+8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7v+8\right)\left(7v+8\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(7v+8\right)^{2}$$

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.

$$factor(49v^{2}+112v+64)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(49,112,64)=1$$

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

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

Kimihia te pūtakerua o te kīanga tau autō, 64.

$$\sqrt{64}=8$$

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.

$$\left(7v+8\right)^{2}$$

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.

$$49v^{2}+112v+64=0$$

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.

$$v=\frac{-112±\sqrt{112^{2}-4\times 49\times 64}}{2\times 49}$$

Pūrua 112.

$$v=\frac{-112±\sqrt{12544-4\times 49\times 64}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$v=\frac{-112±\sqrt{12544-196\times 64}}{2\times 49}$$

Whakareatia -196 ki te 64.

$$v=\frac{-112±\sqrt{12544-12544}}{2\times 49}$$

Tāpiri 12544 ki te -12544.

$$v=\frac{-112±\sqrt{0}}{2\times 49}$$

Tuhia te pūtakerua o te 0.

$$v=\frac{-112±0}{2\times 49}$$

Whakareatia 2 ki te 49.

$$v=\frac{-112±0}{98}$$

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{8}{7} mō te x_{1} me te -\frac{8}{7} mō te x_{2}.

$$49v^{2}+112v+64=49\left(v-\left(-\frac{8}{7}\right)\right)\left(v-\left(-\frac{8}{7}\right)\right)$$

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

$$49v^{2}+112v+64=49\left(v+\frac{8}{7}\right)\left(v+\frac{8}{7}\right)$$

Tāpiri \frac{8}{7} 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.

$$49v^{2}+112v+64=49\times \frac{7v+8}{7}\left(v+\frac{8}{7}\right)$$

Tāpiri \frac{8}{7} 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.

$$49v^{2}+112v+64=49\times \frac{7v+8}{7}\times \frac{7v+8}{7}$$

Whakareatia \frac{7v+8}{7} ki te \frac{7v+8}{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.

$$49v^{2}+112v+64=49\times \frac{\left(7v+8\right)\left(7v+8\right)}{7\times 7}$$

Whakareatia 7 ki te 7.

$$49v^{2}+112v+64=49\times \frac{\left(7v+8\right)\left(7v+8\right)}{49}$$

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

$$49v^{2}+112v+64=\left(7v+8\right)\left(7v+8\right)$$

Aromātai

$$\left(7v+8\right)^{2}$$