Whakaoti i te 20x^2-30x-20

$$20 x ^ { 2 } - 30 x - 20$$

Tauwehe

$$10\left(x-2\right)\left(2x+1\right)$$

Ngā Upane Otinga

Tauwehea te 10.

$$10\left(2x^{2}-3x-2\right)$$

Whakaarohia te 2x^{2}-3x-2. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=2\left(-2\right)=-4$$

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 -4.

$$1,-4 2,-2$$

Tātaihia te tapeke mō ia takirua.

$$1-4=-3 2-2=0$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-4 b=1$$

Tuhia anō te 2x^{2}-3x-2 hei \left(2x^{2}-4x\right)+\left(x-2\right).

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

Whakatauwehea atu 2x i te 2x^{2}-4x.

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

$$10\left(x-2\right)\left(2x+1\right)$$

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.

$$20x^{2}-30x-20=0$$

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.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 20\left(-20\right)}}{2\times 20}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 20\left(-20\right)}}{2\times 20}$$

Whakareatia -4 ki te 20.

$$x=\frac{-\left(-30\right)±\sqrt{900-80\left(-20\right)}}{2\times 20}$$

Whakareatia -80 ki te -20.

$$x=\frac{-\left(-30\right)±\sqrt{900+1600}}{2\times 20}$$

Tāpiri 900 ki te 1600.

$$x=\frac{-\left(-30\right)±\sqrt{2500}}{2\times 20}$$

Tuhia te pūtakerua o te 2500.

$$x=\frac{-\left(-30\right)±50}{2\times 20}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±50}{2\times 20}$$

Whakareatia 2 ki te 20.

$$x=\frac{30±50}{40}$$

Nā, me whakaoti te whārite x=\frac{30±50}{40} ina he tāpiri te ±. Tāpiri 30 ki te 50.

$$x=\frac{80}{40}$$

Whakawehe 80 ki te 40.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{30±50}{40} ina he tango te ±. Tango 50 mai i 30.

$$x=-\frac{20}{40}$$

Whakahekea te hautanga \frac{-20}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

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

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

$$20x^{2}-30x-20=20\left(x-2\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$20x^{2}-30x-20=20\left(x-2\right)\left(x+\frac{1}{2}\right)$$

Tāpiri \frac{1}{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.

$$20x^{2}-30x-20=20\left(x-2\right)\times \frac{2x+1}{2}$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 20 me te 2.

$$20x^{2}-30x-20=10\left(x-2\right)\left(2x+1\right)$$

Aromātai

$$10\left(x-2\right)\left(2x+1\right)$$