Whakaoti i te 6x^2-x-1

$$6 { x }^{ 2 } -x-1$$

Tauwehe

$$\left(2x-1\right)\left(3x+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6x^{2}+ax+bx-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=6\left(-1\right)=-6$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -6.

$$1,-6 2,-3$$

Tātaihia te tapeke mō ia takirua.

$$1-6=-5 2-3=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-3 b=2$$

Tuhia anō te 6x^{2}-x-1 hei \left(6x^{2}-3x\right)+\left(2x-1\right).

$$\left(6x^{2}-3x\right)+\left(2x-1\right)$$

Whakatauwehea atu 3x i te 6x^{2}-3x.

$$3x\left(2x-1\right)+2x-1$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(3x+1\right)$$

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.

$$6x^{2}-x-1=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 6\left(-1\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-1\right)±\sqrt{1-24\left(-1\right)}}{2\times 6}$$

Whakareatia -24 ki te -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+24}}{2\times 6}$$

Tāpiri 1 ki te 24.

$$x=\frac{-\left(-1\right)±\sqrt{25}}{2\times 6}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-\left(-1\right)±5}{2\times 6}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±5}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{1±5}{12}$$

Nā, me whakaoti te whārite x=\frac{1±5}{12} ina he tāpiri te ±. Tāpiri 1 ki te 5.

$$x=\frac{6}{12}$$

Whakahekea te hautanga \frac{6}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{1±5}{12} ina he tango te ±. Tango 5 mai i 1.

$$x=-\frac{4}{12}$$

Whakahekea te hautanga \frac{-4}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{1}{3}$$

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

$$6x^{2}-x-1=6\left(x-\frac{1}{2}\right)\left(x-\left(-\frac{1}{3}\right)\right)$$

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

$$6x^{2}-x-1=6\left(x-\frac{1}{2}\right)\left(x+\frac{1}{3}\right)$$

Tango \frac{1}{2} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$6x^{2}-x-1=6\times \frac{2x-1}{2}\left(x+\frac{1}{3}\right)$$

Tāpiri \frac{1}{3} 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.

$$6x^{2}-x-1=6\times \frac{2x-1}{2}\times \frac{3x+1}{3}$$

Whakareatia \frac{2x-1}{2} ki te \frac{3x+1}{3} 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.

$$6x^{2}-x-1=6\times \frac{\left(2x-1\right)\left(3x+1\right)}{2\times 3}$$

Whakareatia 2 ki te 3.

$$6x^{2}-x-1=6\times \frac{\left(2x-1\right)\left(3x+1\right)}{6}$$

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

$$6x^{2}-x-1=\left(2x-1\right)\left(3x+1\right)$$

Aromātai

$$\left(2x-1\right)\left(3x+1\right)$$