Whakaoti i te 4d^2+36d+81

$$4 d ^ { 2 } + 36 d + 81$$

Tauwehe

$$\left(2d+9\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4d^{2}+ad+bd+81. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=36 ab=4\times 81=324$$

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 324.

$$1,324 2,162 3,108 4,81 6,54 9,36 12,27 18,18$$

Tātaihia te tapeke mō ia takirua.

$$1+324=325 2+162=164 3+108=111 4+81=85 6+54=60 9+36=45 12+27=39 18+18=36$$

Ko te otinga te takirua ka hoatu i te tapeke 36.

$$a=18 b=18$$

Tuhia anō te 4d^{2}+36d+81 hei \left(4d^{2}+18d\right)+\left(18d+81\right).

$$\left(4d^{2}+18d\right)+\left(18d+81\right)$$

Tauwehea te 2d i te tuatahi me te 9 i te rōpū tuarua.

$$2d\left(2d+9\right)+9\left(2d+9\right)$$

Whakatauwehea atu te kīanga pātahi 2d+9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2d+9\right)\left(2d+9\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(2d+9\right)^{2}$$

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.

$$factor(4d^{2}+36d+81)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(4,36,81)=1$$

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

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

Kimihia te pūtakerua o te kīanga tau autō, 81.

$$\sqrt{81}=9$$

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.

$$\left(2d+9\right)^{2}$$

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.

$$4d^{2}+36d+81=0$$

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.

$$d=\frac{-36±\sqrt{36^{2}-4\times 4\times 81}}{2\times 4}$$

Pūrua 36.

$$d=\frac{-36±\sqrt{1296-4\times 4\times 81}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$d=\frac{-36±\sqrt{1296-16\times 81}}{2\times 4}$$

Whakareatia -16 ki te 81.

$$d=\frac{-36±\sqrt{1296-1296}}{2\times 4}$$

Tāpiri 1296 ki te -1296.

$$d=\frac{-36±\sqrt{0}}{2\times 4}$$

Tuhia te pūtakerua o te 0.

$$d=\frac{-36±0}{2\times 4}$$

Whakareatia 2 ki te 4.

$$d=\frac{-36±0}{8}$$

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{9}{2} mō te x_{1} me te -\frac{9}{2} mō te x_{2}.

$$4d^{2}+36d+81=4\left(d-\left(-\frac{9}{2}\right)\right)\left(d-\left(-\frac{9}{2}\right)\right)$$

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

$$4d^{2}+36d+81=4\left(d+\frac{9}{2}\right)\left(d+\frac{9}{2}\right)$$

Tāpiri \frac{9}{2} ki te d 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.

$$4d^{2}+36d+81=4\times \frac{2d+9}{2}\left(d+\frac{9}{2}\right)$$

Tāpiri \frac{9}{2} ki te d 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.

$$4d^{2}+36d+81=4\times \frac{2d+9}{2}\times \frac{2d+9}{2}$$

Whakareatia \frac{2d+9}{2} ki te \frac{2d+9}{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.

$$4d^{2}+36d+81=4\times \frac{\left(2d+9\right)\left(2d+9\right)}{2\times 2}$$

Whakareatia 2 ki te 2.

$$4d^{2}+36d+81=4\times \frac{\left(2d+9\right)\left(2d+9\right)}{4}$$

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

$$4d^{2}+36d+81=\left(2d+9\right)\left(2d+9\right)$$

Aromātai

$$\left(2d+9\right)^{2}$$