Whakaoti i te 2x^2-35x-18

$$2 { x }^{ 2 } -35x-18$$

Tauwehe

$$\left(x-18\right)\left(2x+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2x^{2}+ax+bx-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-35 ab=2\left(-18\right)=-36$$

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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Ko te otinga te takirua ka hoatu i te tapeke -35.

$$a=-36 b=1$$

Tuhia anō te 2x^{2}-35x-18 hei \left(2x^{2}-36x\right)+\left(x-18\right).

$$\left(2x^{2}-36x\right)+\left(x-18\right)$$

Whakatauwehea atu 2x i te 2x^{2}-36x.

$$2x\left(x-18\right)+x-18$$

Whakatauwehea atu te kīanga pātahi x-18 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-18\right)\left(2x+1\right)$$

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.

$$2x^{2}-35x-18=0$$

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.

$$x=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

Pūrua -35.

$$x=\frac{-\left(-35\right)±\sqrt{1225-4\times 2\left(-18\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-35\right)±\sqrt{1225-8\left(-18\right)}}{2\times 2}$$

Whakareatia -8 ki te -18.

$$x=\frac{-\left(-35\right)±\sqrt{1225+144}}{2\times 2}$$

Tāpiri 1225 ki te 144.

$$x=\frac{-\left(-35\right)±\sqrt{1369}}{2\times 2}$$

Tuhia te pūtakerua o te 1369.

$$x=\frac{-\left(-35\right)±37}{2\times 2}$$

Ko te tauaro o -35 ko 35.

$$x=\frac{35±37}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{35±37}{4}$$

Nā, me whakaoti te whārite x=\frac{35±37}{4} ina he tāpiri te ±. Tāpiri 35 ki te 37.

$$x=\frac{72}{4}$$

Whakawehe 72 ki te 4.

$$x=18$$

Nā, me whakaoti te whārite x=\frac{35±37}{4} ina he tango te ±. Tango 37 mai i 35.

$$x=-\frac{2}{4}$$

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

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

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

$$2x^{2}-35x-18=2\left(x-18\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$2x^{2}-35x-18=2\left(x-18\right)\left(x+\frac{1}{2}\right)$$

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.

$$2x^{2}-35x-18=2\left(x-18\right)\times \frac{2x+1}{2}$$

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

$$2x^{2}-35x-18=\left(x-18\right)\left(2x+1\right)$$

Aromātai

$$\left(x-18\right)\left(2x+1\right)$$