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

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

Tauwehe

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

Ngā Upane Otinga

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

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

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 -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.

$$a=-14 b=3$$

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

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

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

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

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

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

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.

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

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.

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

Pūrua -11.

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

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-11\right)±\sqrt{121-8\left(-21\right)}}{2\times 2}$$

Whakareatia -8 ki te -21.

$$x=\frac{-\left(-11\right)±\sqrt{121+168}}{2\times 2}$$

Tāpiri 121 ki te 168.

$$x=\frac{-\left(-11\right)±\sqrt{289}}{2\times 2}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-\left(-11\right)±17}{2\times 2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±17}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{11±17}{4}$$

Nā, me whakaoti te whārite x=\frac{11±17}{4} ina he tāpiri te ±. Tāpiri 11 ki te 17.

$$x=\frac{28}{4}$$

Whakawehe 28 ki te 4.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{11±17}{4} ina he tango te ±. Tango 17 mai i 11.

$$x=-\frac{6}{4}$$

Whakahekea te hautanga \frac{-6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

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

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

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

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

Tāpiri \frac{3}{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.

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

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 2 me te 2.

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

Aromātai

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