Whakaoti i te 6w^2+55w+9

$$6 w ^ { 2 } + 55 w + 9$$

Tauwehe

$$\left(w+9\right)\left(6w+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6w^{2}+aw+bw+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=55 ab=6\times 9=54$$

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 54.

$$1,54 2,27 3,18 6,9$$

Tātaihia te tapeke mō ia takirua.

$$1+54=55 2+27=29 3+18=21 6+9=15$$

Ko te otinga te takirua ka hoatu i te tapeke 55.

$$a=1 b=54$$

Tuhia anō te 6w^{2}+55w+9 hei \left(6w^{2}+w\right)+\left(54w+9\right).

$$\left(6w^{2}+w\right)+\left(54w+9\right)$$

Tauwehea te w i te tuatahi me te 9 i te rōpū tuarua.

$$w\left(6w+1\right)+9\left(6w+1\right)$$

Whakatauwehea atu te kīanga pātahi 6w+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6w+1\right)\left(w+9\right)$$

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.

$$6w^{2}+55w+9=0$$

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.

$$w=\frac{-55±\sqrt{55^{2}-4\times 6\times 9}}{2\times 6}$$

Pūrua 55.

$$w=\frac{-55±\sqrt{3025-4\times 6\times 9}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$w=\frac{-55±\sqrt{3025-24\times 9}}{2\times 6}$$

Whakareatia -24 ki te 9.

$$w=\frac{-55±\sqrt{3025-216}}{2\times 6}$$

Tāpiri 3025 ki te -216.

$$w=\frac{-55±\sqrt{2809}}{2\times 6}$$

Tuhia te pūtakerua o te 2809.

$$w=\frac{-55±53}{2\times 6}$$

Whakareatia 2 ki te 6.

$$w=\frac{-55±53}{12}$$

Nā, me whakaoti te whārite w=\frac{-55±53}{12} ina he tāpiri te ±. Tāpiri -55 ki te 53.

$$w=-\frac{2}{12}$$

Whakahekea te hautanga \frac{-2}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$w=-\frac{1}{6}$$

Nā, me whakaoti te whārite w=\frac{-55±53}{12} ina he tango te ±. Tango 53 mai i -55.

$$w=-\frac{108}{12}$$

Whakawehe -108 ki te 12.

$$w=-9$$

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{1}{6} mō te x_{1} me te -9 mō te x_{2}.

$$6w^{2}+55w+9=6\left(w-\left(-\frac{1}{6}\right)\right)\left(w-\left(-9\right)\right)$$

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

$$6w^{2}+55w+9=6\left(w+\frac{1}{6}\right)\left(w+9\right)$$

Tāpiri \frac{1}{6} ki te w 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.

$$6w^{2}+55w+9=6\times \frac{6w+1}{6}\left(w+9\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 6 i roto i te 6 me te 6.

$$6w^{2}+55w+9=\left(6w+1\right)\left(w+9\right)$$

Aromātai

$$\left(w+9\right)\left(6w+1\right)$$