Whakaoti i te 9+12v+4v^2

$$9 + 12 v + 4 v ^ { 2 }$$

Tauwehe

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

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.

$$4v^{2}+12v+9$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4v^{2}+av+bv+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=4\times 9=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Tātaihia te tapeke mō ia takirua.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

Ko te otinga te takirua ka hoatu i te tapeke 12.

$$a=6 b=6$$

Tuhia anō te 4v^{2}+12v+9 hei \left(4v^{2}+6v\right)+\left(6v+9\right).

$$\left(4v^{2}+6v\right)+\left(6v+9\right)$$

Tauwehea te 2v i te tuatahi me te 3 i te rōpū tuarua.

$$2v\left(2v+3\right)+3\left(2v+3\right)$$

Whakatauwehea atu te kīanga pātahi 2v+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2v+3\right)\left(2v+3\right)$$

Tuhia anōtia hei pūrua huarua.

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

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.

$$factor(4v^{2}+12v+9)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(4,12,9)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 4v^{2}.

$$\sqrt{4v^{2}}=2v$$

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.

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

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.

$$4v^{2}+12v+9=0$$

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.

$$v=\frac{-12±\sqrt{12^{2}-4\times 4\times 9}}{2\times 4}$$

Pūrua 12.

$$v=\frac{-12±\sqrt{144-4\times 4\times 9}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$v=\frac{-12±\sqrt{144-16\times 9}}{2\times 4}$$

Whakareatia -16 ki te 9.

$$v=\frac{-12±\sqrt{144-144}}{2\times 4}$$

Tāpiri 144 ki te -144.

$$v=\frac{-12±\sqrt{0}}{2\times 4}$$

Tuhia te pūtakerua o te 0.

$$v=\frac{-12±0}{2\times 4}$$

Whakareatia 2 ki te 4.

$$v=\frac{-12±0}{8}$$

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

$$4v^{2}+12v+9=4\left(v-\left(-\frac{3}{2}\right)\right)\left(v-\left(-\frac{3}{2}\right)\right)$$

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

$$4v^{2}+12v+9=4\left(v+\frac{3}{2}\right)\left(v+\frac{3}{2}\right)$$

Tāpiri \frac{3}{2} 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.

$$4v^{2}+12v+9=4\times \frac{2v+3}{2}\left(v+\frac{3}{2}\right)$$

Tāpiri \frac{3}{2} 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.

$$4v^{2}+12v+9=4\times \frac{2v+3}{2}\times \frac{2v+3}{2}$$

Whakareatia \frac{2v+3}{2} ki te \frac{2v+3}{2} 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.

$$4v^{2}+12v+9=4\times \frac{\left(2v+3\right)\left(2v+3\right)}{2\times 2}$$

Whakareatia 2 ki te 2.

$$4v^{2}+12v+9=4\times \frac{\left(2v+3\right)\left(2v+3\right)}{4}$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te 4 me te 4.

$$4v^{2}+12v+9=\left(2v+3\right)\left(2v+3\right)$$

Aromātai

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