Whakaoti i te 4x^2-21x-18

$$4 { x }^{ 2 } -21x-18$$

Tauwehe

$$\left(x-6\right)\left(4x+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4x^{2}+ax+bx-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-21 ab=4\left(-18\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -21.

$$a=-24 b=3$$

Tuhia anō te 4x^{2}-21x-18 hei \left(4x^{2}-24x\right)+\left(3x-18\right).

$$\left(4x^{2}-24x\right)+\left(3x-18\right)$$

Tauwehea te 4x i te tuatahi me te 3 i te rōpū tuarua.

$$4x\left(x-6\right)+3\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(4x+3\right)$$

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.

$$4x^{2}-21x-18=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 4\left(-18\right)}}{2\times 4}$$

Pūrua -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 4\left(-18\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-21\right)±\sqrt{441-16\left(-18\right)}}{2\times 4}$$

Whakareatia -16 ki te -18.

$$x=\frac{-\left(-21\right)±\sqrt{441+288}}{2\times 4}$$

Tāpiri 441 ki te 288.

$$x=\frac{-\left(-21\right)±\sqrt{729}}{2\times 4}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-\left(-21\right)±27}{2\times 4}$$

Ko te tauaro o -21 ko 21.

$$x=\frac{21±27}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{21±27}{8}$$

Nā, me whakaoti te whārite x=\frac{21±27}{8} ina he tāpiri te ±. Tāpiri 21 ki te 27.

$$x=\frac{48}{8}$$

Whakawehe 48 ki te 8.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{21±27}{8} ina he tango te ±. Tango 27 mai i 21.

$$x=-\frac{6}{8}$$

Whakahekea te hautanga \frac{-6}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{3}{4}$$

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 6 mō te x_{1} me te -\frac{3}{4} mō te x_{2}.

$$4x^{2}-21x-18=4\left(x-6\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$4x^{2}-21x-18=4\left(x-6\right)\left(x+\frac{3}{4}\right)$$

Tāpiri \frac{3}{4} 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.

$$4x^{2}-21x-18=4\left(x-6\right)\times \frac{4x+3}{4}$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te 4 me te 4.

$$4x^{2}-21x-18=\left(x-6\right)\left(4x+3\right)$$

Aromātai

$$\left(x-6\right)\left(4x+3\right)$$