Whakaoti i te 3n^2-357n+10620

$$3 n ^ { 2 } - 357 n + 10620$$

Tauwehe

$$3\left(n-60\right)\left(n-59\right)$$

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(n^{2}-119n+3540\right)$$

Whakaarohia te n^{2}-119n+3540. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei n^{2}+an+bn+3540. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-119 ab=1\times 3540=3540$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 3540.

$$-1,-3540 -2,-1770 -3,-1180 -4,-885 -5,-708 -6,-590 -10,-354 -12,-295 -15,-236 -20,-177 -30,-118 -59,-60$$

Tātaihia te tapeke mō ia takirua.

$$-1-3540=-3541 -2-1770=-1772 -3-1180=-1183 -4-885=-889 -5-708=-713 -6-590=-596 -10-354=-364 -12-295=-307 -15-236=-251 -20-177=-197 -30-118=-148 -59-60=-119$$

Ko te otinga te takirua ka hoatu i te tapeke -119.

$$a=-60 b=-59$$

Tuhia anō te n^{2}-119n+3540 hei \left(n^{2}-60n\right)+\left(-59n+3540\right).

$$\left(n^{2}-60n\right)+\left(-59n+3540\right)$$

Tauwehea te n i te tuatahi me te -59 i te rōpū tuarua.

$$n\left(n-60\right)-59\left(n-60\right)$$

Whakatauwehea atu te kīanga pātahi n-60 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-60\right)\left(n-59\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$3\left(n-60\right)\left(n-59\right)$$

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.

$$3n^{2}-357n+10620=0$$

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.

$$n=\frac{-\left(-357\right)±\sqrt{\left(-357\right)^{2}-4\times 3\times 10620}}{2\times 3}$$

Pūrua -357.

$$n=\frac{-\left(-357\right)±\sqrt{127449-4\times 3\times 10620}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$n=\frac{-\left(-357\right)±\sqrt{127449-12\times 10620}}{2\times 3}$$

Whakareatia -12 ki te 10620.

$$n=\frac{-\left(-357\right)±\sqrt{127449-127440}}{2\times 3}$$

Tāpiri 127449 ki te -127440.

$$n=\frac{-\left(-357\right)±\sqrt{9}}{2\times 3}$$

Tuhia te pūtakerua o te 9.

$$n=\frac{-\left(-357\right)±3}{2\times 3}$$

Ko te tauaro o -357 ko 357.

$$n=\frac{357±3}{2\times 3}$$

Whakareatia 2 ki te 3.

$$n=\frac{357±3}{6}$$

Nā, me whakaoti te whārite n=\frac{357±3}{6} ina he tāpiri te ±. Tāpiri 357 ki te 3.

$$n=\frac{360}{6}$$

Whakawehe 360 ki te 6.

$$n=60$$

Nā, me whakaoti te whārite n=\frac{357±3}{6} ina he tango te ±. Tango 3 mai i 357.

$$n=\frac{354}{6}$$

Whakawehe 354 ki te 6.

$$n=59$$

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 60 mō te x_{1} me te 59 mō te x_{2}.

$$3n^{2}-357n+10620=3\left(n-60\right)\left(n-59\right)$$

Aromātai

$$3\left(n-60\right)\left(n-59\right)$$