Whakaoti i te 21x^2+11x-2

$$21 x ^ { 2 } + 11 x - 2$$

Tauwehe

$$\left(7x-1\right)\left(3x+2\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 21x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=21\left(-2\right)=-42$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -42.

$$-1,42 -2,21 -3,14 -6,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=-3 b=14$$

Tuhia anō te 21x^{2}+11x-2 hei \left(21x^{2}-3x\right)+\left(14x-2\right).

$$\left(21x^{2}-3x\right)+\left(14x-2\right)$$

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

$$3x\left(7x-1\right)+2\left(7x-1\right)$$

Whakatauwehea atu te kīanga pātahi 7x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-1\right)\left(3x+2\right)$$

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.

$$21x^{2}+11x-2=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 21\left(-2\right)}}{2\times 21}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\times 21\left(-2\right)}}{2\times 21}$$

Whakareatia -4 ki te 21.

$$x=\frac{-11±\sqrt{121-84\left(-2\right)}}{2\times 21}$$

Whakareatia -84 ki te -2.

$$x=\frac{-11±\sqrt{121+168}}{2\times 21}$$

Tāpiri 121 ki te 168.

$$x=\frac{-11±\sqrt{289}}{2\times 21}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-11±17}{2\times 21}$$

Whakareatia 2 ki te 21.

$$x=\frac{-11±17}{42}$$

Nā, me whakaoti te whārite x=\frac{-11±17}{42} ina he tāpiri te ±. Tāpiri -11 ki te 17.

$$x=\frac{6}{42}$$

Whakahekea te hautanga \frac{6}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{1}{7}$$

Nā, me whakaoti te whārite x=\frac{-11±17}{42} ina he tango te ±. Tango 17 mai i -11.

$$x=-\frac{28}{42}$$

Whakahekea te hautanga \frac{-28}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$x=-\frac{2}{3}$$

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

$$21x^{2}+11x-2=21\left(x-\frac{1}{7}\right)\left(x-\left(-\frac{2}{3}\right)\right)$$

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

$$21x^{2}+11x-2=21\left(x-\frac{1}{7}\right)\left(x+\frac{2}{3}\right)$$

Tango \frac{1}{7} 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.

$$21x^{2}+11x-2=21\times \frac{7x-1}{7}\left(x+\frac{2}{3}\right)$$

Tāpiri \frac{2}{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.

$$21x^{2}+11x-2=21\times \frac{7x-1}{7}\times \frac{3x+2}{3}$$

Whakareatia \frac{7x-1}{7} ki te \frac{3x+2}{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.

$$21x^{2}+11x-2=21\times \frac{\left(7x-1\right)\left(3x+2\right)}{7\times 3}$$

Whakareatia 7 ki te 3.

$$21x^{2}+11x-2=21\times \frac{\left(7x-1\right)\left(3x+2\right)}{21}$$

Whakakorea atu te tauwehe pūnoa nui rawa 21 i roto i te 21 me te 21.

$$21x^{2}+11x-2=\left(7x-1\right)\left(3x+2\right)$$

Aromātai

$$\left(7x-1\right)\left(3x+2\right)$$