Whakaoti i te 6x^2+47x+35

$$6 x ^ { 2 } + 47 x + 35$$

Tauwehe

$$\left(x+7\right)\left(6x+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6x^{2}+ax+bx+35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=47 ab=6\times 35=210$$

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 210.

$$1,210 2,105 3,70 5,42 6,35 7,30 10,21 14,15$$

Tātaihia te tapeke mō ia takirua.

$$1+210=211 2+105=107 3+70=73 5+42=47 6+35=41 7+30=37 10+21=31 14+15=29$$

Ko te otinga te takirua ka hoatu i te tapeke 47.

$$a=5 b=42$$

Tuhia anō te 6x^{2}+47x+35 hei \left(6x^{2}+5x\right)+\left(42x+35\right).

$$\left(6x^{2}+5x\right)+\left(42x+35\right)$$

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(6x+5\right)+7\left(6x+5\right)$$

Whakatauwehea atu te kīanga pātahi 6x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6x+5\right)\left(x+7\right)$$

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.

$$6x^{2}+47x+35=0$$

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.

$$x=\frac{-47±\sqrt{47^{2}-4\times 6\times 35}}{2\times 6}$$

Pūrua 47.

$$x=\frac{-47±\sqrt{2209-4\times 6\times 35}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-47±\sqrt{2209-24\times 35}}{2\times 6}$$

Whakareatia -24 ki te 35.

$$x=\frac{-47±\sqrt{2209-840}}{2\times 6}$$

Tāpiri 2209 ki te -840.

$$x=\frac{-47±\sqrt{1369}}{2\times 6}$$

Tuhia te pūtakerua o te 1369.

$$x=\frac{-47±37}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-47±37}{12}$$

Nā, me whakaoti te whārite x=\frac{-47±37}{12} ina he tāpiri te ±. Tāpiri -47 ki te 37.

$$x=-\frac{10}{12}$$

Whakahekea te hautanga \frac{-10}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{5}{6}$$

Nā, me whakaoti te whārite x=\frac{-47±37}{12} ina he tango te ±. Tango 37 mai i -47.

$$x=-\frac{84}{12}$$

Whakawehe -84 ki te 12.

$$x=-7$$

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{5}{6} mō te x_{1} me te -7 mō te x_{2}.

$$6x^{2}+47x+35=6\left(x-\left(-\frac{5}{6}\right)\right)\left(x-\left(-7\right)\right)$$

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

$$6x^{2}+47x+35=6\left(x+\frac{5}{6}\right)\left(x+7\right)$$

Tāpiri \frac{5}{6} ki te x 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.

$$6x^{2}+47x+35=6\times \frac{6x+5}{6}\left(x+7\right)$$

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

$$6x^{2}+47x+35=\left(6x+5\right)\left(x+7\right)$$

Aromātai

$$\left(x+7\right)\left(6x+5\right)$$