Whakaoti i te 6x^2+11x-10

$$6 x ^ { 2 } + 11 x - 10$$

Tauwehe

$$\left(3x-2\right)\left(2x+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6x^{2}+ax+bx-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=6\left(-10\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

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

$$a=-4 b=15$$

Tuhia anō te 6x^{2}+11x-10 hei \left(6x^{2}-4x\right)+\left(15x-10\right).

$$\left(6x^{2}-4x\right)+\left(15x-10\right)$$

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(3x-2\right)+5\left(3x-2\right)$$

Whakatauwehea atu te kīanga pātahi 3x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-2\right)\left(2x+5\right)$$

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.

$$6x^{2}+11x-10=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 6\left(-10\right)}}{2\times 6}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\times 6\left(-10\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-11±\sqrt{121-24\left(-10\right)}}{2\times 6}$$

Whakareatia -24 ki te -10.

$$x=\frac{-11±\sqrt{121+240}}{2\times 6}$$

Tāpiri 121 ki te 240.

$$x=\frac{-11±\sqrt{361}}{2\times 6}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-11±19}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-11±19}{12}$$

Nā, me whakaoti te whārite x=\frac{-11±19}{12} ina he tāpiri te ±. Tāpiri -11 ki te 19.

$$x=\frac{8}{12}$$

Whakahekea te hautanga \frac{8}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite x=\frac{-11±19}{12} ina he tango te ±. Tango 19 mai i -11.

$$x=-\frac{30}{12}$$

Whakahekea te hautanga \frac{-30}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{5}{2}$$

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{2}{3} mō te x_{1} me te -\frac{5}{2} mō te x_{2}.

$$6x^{2}+11x-10=6\left(x-\frac{2}{3}\right)\left(x-\left(-\frac{5}{2}\right)\right)$$

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

$$6x^{2}+11x-10=6\left(x-\frac{2}{3}\right)\left(x+\frac{5}{2}\right)$$

Tango \frac{2}{3} 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.

$$6x^{2}+11x-10=6\times \frac{3x-2}{3}\left(x+\frac{5}{2}\right)$$

Tāpiri \frac{5}{2} 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.

$$6x^{2}+11x-10=6\times \frac{3x-2}{3}\times \frac{2x+5}{2}$$

Whakareatia \frac{3x-2}{3} ki te \frac{2x+5}{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.

$$6x^{2}+11x-10=6\times \frac{\left(3x-2\right)\left(2x+5\right)}{3\times 2}$$

Whakareatia 3 ki te 2.

$$6x^{2}+11x-10=6\times \frac{\left(3x-2\right)\left(2x+5\right)}{6}$$

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

$$6x^{2}+11x-10=\left(3x-2\right)\left(2x+5\right)$$

Aromātai

$$\left(3x-2\right)\left(2x+5\right)$$