Whakaoti i te 24n^2-65n+21

$$24 n ^ { 2 } - 65 n + 21$$

Tauwehe

$$\left(3n-7\right)\left(8n-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 24n^{2}+an+bn+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-65 ab=24\times 21=504$$

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 504.

$$-1,-504 -2,-252 -3,-168 -4,-126 -6,-84 -7,-72 -8,-63 -9,-56 -12,-42 -14,-36 -18,-28 -21,-24$$

Tātaihia te tapeke mō ia takirua.

$$-1-504=-505 -2-252=-254 -3-168=-171 -4-126=-130 -6-84=-90 -7-72=-79 -8-63=-71 -9-56=-65 -12-42=-54 -14-36=-50 -18-28=-46 -21-24=-45$$

Ko te otinga te takirua ka hoatu i te tapeke -65.

$$a=-56 b=-9$$

Tuhia anō te 24n^{2}-65n+21 hei \left(24n^{2}-56n\right)+\left(-9n+21\right).

$$\left(24n^{2}-56n\right)+\left(-9n+21\right)$$

Tauwehea te 8n i te tuatahi me te -3 i te rōpū tuarua.

$$8n\left(3n-7\right)-3\left(3n-7\right)$$

Whakatauwehea atu te kīanga pātahi 3n-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3n-7\right)\left(8n-3\right)$$

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.

$$24n^{2}-65n+21=0$$

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.

$$n=\frac{-\left(-65\right)±\sqrt{\left(-65\right)^{2}-4\times 24\times 21}}{2\times 24}$$

Pūrua -65.

$$n=\frac{-\left(-65\right)±\sqrt{4225-4\times 24\times 21}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$n=\frac{-\left(-65\right)±\sqrt{4225-96\times 21}}{2\times 24}$$

Whakareatia -96 ki te 21.

$$n=\frac{-\left(-65\right)±\sqrt{4225-2016}}{2\times 24}$$

Tāpiri 4225 ki te -2016.

$$n=\frac{-\left(-65\right)±\sqrt{2209}}{2\times 24}$$

Tuhia te pūtakerua o te 2209.

$$n=\frac{-\left(-65\right)±47}{2\times 24}$$

Ko te tauaro o -65 ko 65.

$$n=\frac{65±47}{2\times 24}$$

Whakareatia 2 ki te 24.

$$n=\frac{65±47}{48}$$

Nā, me whakaoti te whārite n=\frac{65±47}{48} ina he tāpiri te ±. Tāpiri 65 ki te 47.

$$n=\frac{112}{48}$$

Whakahekea te hautanga \frac{112}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$n=\frac{7}{3}$$

Nā, me whakaoti te whārite n=\frac{65±47}{48} ina he tango te ±. Tango 47 mai i 65.

$$n=\frac{18}{48}$$

Whakahekea te hautanga \frac{18}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$n=\frac{3}{8}$$

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 \frac{7}{3} mō te x_{1} me te \frac{3}{8} mō te x_{2}.

$$24n^{2}-65n+21=24\left(n-\frac{7}{3}\right)\left(n-\frac{3}{8}\right)$$

Tango \frac{7}{3} mai i n mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$24n^{2}-65n+21=24\times \frac{3n-7}{3}\left(n-\frac{3}{8}\right)$$

Tango \frac{3}{8} mai i n mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$24n^{2}-65n+21=24\times \frac{3n-7}{3}\times \frac{8n-3}{8}$$

Whakareatia \frac{3n-7}{3} ki te \frac{8n-3}{8} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$24n^{2}-65n+21=24\times \frac{\left(3n-7\right)\left(8n-3\right)}{3\times 8}$$

Whakareatia 3 ki te 8.

$$24n^{2}-65n+21=24\times \frac{\left(3n-7\right)\left(8n-3\right)}{24}$$

Whakakorea atu te tauwehe pūnoa nui rawa 24 i roto i te 24 me te 24.

$$24n^{2}-65n+21=\left(3n-7\right)\left(8n-3\right)$$

Aromātai

$$\left(3n-7\right)\left(8n-3\right)$$