Whakaoti i te 4n^2-2n-90

$$4 n ^ { 2 } - 2 n - 90$$

Tauwehe

$$2\left(n-5\right)\left(2n+9\right)$$

Ngā Upane Otinga

Tauwehea te 2.

$$2\left(2n^{2}-n-45\right)$$

Whakaarohia te 2n^{2}-n-45. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2n^{2}+an+bn-45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=2\left(-45\right)=-90$$

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 -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-10 b=9$$

Tuhia anō te 2n^{2}-n-45 hei \left(2n^{2}-10n\right)+\left(9n-45\right).

$$\left(2n^{2}-10n\right)+\left(9n-45\right)$$

Tauwehea te 2n i te tuatahi me te 9 i te rōpū tuarua.

$$2n\left(n-5\right)+9\left(n-5\right)$$

Whakatauwehea atu te kīanga pātahi n-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-5\right)\left(2n+9\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$2\left(n-5\right)\left(2n+9\right)$$

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.

$$4n^{2}-2n-90=0$$

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.

$$n=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 4\left(-90\right)}}{2\times 4}$$

Pūrua -2.

$$n=\frac{-\left(-2\right)±\sqrt{4-4\times 4\left(-90\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$n=\frac{-\left(-2\right)±\sqrt{4-16\left(-90\right)}}{2\times 4}$$

Whakareatia -16 ki te -90.

$$n=\frac{-\left(-2\right)±\sqrt{4+1440}}{2\times 4}$$

Tāpiri 4 ki te 1440.

$$n=\frac{-\left(-2\right)±\sqrt{1444}}{2\times 4}$$

Tuhia te pūtakerua o te 1444.

$$n=\frac{-\left(-2\right)±38}{2\times 4}$$

Ko te tauaro o -2 ko 2.

$$n=\frac{2±38}{2\times 4}$$

Whakareatia 2 ki te 4.

$$n=\frac{2±38}{8}$$

Nā, me whakaoti te whārite n=\frac{2±38}{8} ina he tāpiri te ±. Tāpiri 2 ki te 38.

$$n=\frac{40}{8}$$

Whakawehe 40 ki te 8.

$$n=5$$

Nā, me whakaoti te whārite n=\frac{2±38}{8} ina he tango te ±. Tango 38 mai i 2.

$$n=-\frac{36}{8}$$

Whakahekea te hautanga \frac{-36}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$n=-\frac{9}{2}$$

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 5 mō te x_{1} me te -\frac{9}{2} mō te x_{2}.

$$4n^{2}-2n-90=4\left(n-5\right)\left(n-\left(-\frac{9}{2}\right)\right)$$

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

$$4n^{2}-2n-90=4\left(n-5\right)\left(n+\frac{9}{2}\right)$$

Tāpiri \frac{9}{2} ki te n 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.

$$4n^{2}-2n-90=4\left(n-5\right)\times \frac{2n+9}{2}$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 4 me te 2.

$$4n^{2}-2n-90=2\left(n-5\right)\left(2n+9\right)$$

Aromātai

$$2\left(n-5\right)\left(2n+9\right)$$