Whakaoti i te 25a^2-35a+12

$$25 a ^ { 2 } - 35 a + 12$$

Tauwehe

$$\left(5a-4\right)\left(5a-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 25a^{2}+pa+qa+12. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-35 pq=25\times 12=300$$

I te mea kua tōrunga te pq, he ōrite te tohu o p me q. I te mea kua tōraro te p+q, he tōraro hoki a p me q. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 300.

$$-1,-300 -2,-150 -3,-100 -4,-75 -5,-60 -6,-50 -10,-30 -12,-25 -15,-20$$

Tātaihia te tapeke mō ia takirua.

$$-1-300=-301 -2-150=-152 -3-100=-103 -4-75=-79 -5-60=-65 -6-50=-56 -10-30=-40 -12-25=-37 -15-20=-35$$

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

$$p=-20 q=-15$$

Tuhia anō te 25a^{2}-35a+12 hei \left(25a^{2}-20a\right)+\left(-15a+12\right).

$$\left(25a^{2}-20a\right)+\left(-15a+12\right)$$

Tauwehea te 5a i te tuatahi me te -3 i te rōpū tuarua.

$$5a\left(5a-4\right)-3\left(5a-4\right)$$

Whakatauwehea atu te kīanga pātahi 5a-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5a-4\right)\left(5a-3\right)$$

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.

$$25a^{2}-35a+12=0$$

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.

$$a=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\times 25\times 12}}{2\times 25}$$

Pūrua -35.

$$a=\frac{-\left(-35\right)±\sqrt{1225-4\times 25\times 12}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$a=\frac{-\left(-35\right)±\sqrt{1225-100\times 12}}{2\times 25}$$

Whakareatia -100 ki te 12.

$$a=\frac{-\left(-35\right)±\sqrt{1225-1200}}{2\times 25}$$

Tāpiri 1225 ki te -1200.

$$a=\frac{-\left(-35\right)±\sqrt{25}}{2\times 25}$$

Tuhia te pūtakerua o te 25.

$$a=\frac{-\left(-35\right)±5}{2\times 25}$$

Ko te tauaro o -35 ko 35.

$$a=\frac{35±5}{2\times 25}$$

Whakareatia 2 ki te 25.

$$a=\frac{35±5}{50}$$

Nā, me whakaoti te whārite a=\frac{35±5}{50} ina he tāpiri te ±. Tāpiri 35 ki te 5.

$$a=\frac{40}{50}$$

Whakahekea te hautanga \frac{40}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$a=\frac{4}{5}$$

Nā, me whakaoti te whārite a=\frac{35±5}{50} ina he tango te ±. Tango 5 mai i 35.

$$a=\frac{30}{50}$$

Whakahekea te hautanga \frac{30}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$a=\frac{3}{5}$$

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{4}{5} mō te x_{1} me te \frac{3}{5} mō te x_{2}.

$$25a^{2}-35a+12=25\left(a-\frac{4}{5}\right)\left(a-\frac{3}{5}\right)$$

Tango \frac{4}{5} mai i a 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.

$$25a^{2}-35a+12=25\times \frac{5a-4}{5}\left(a-\frac{3}{5}\right)$$

Tango \frac{3}{5} mai i a 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.

$$25a^{2}-35a+12=25\times \frac{5a-4}{5}\times \frac{5a-3}{5}$$

Whakareatia \frac{5a-4}{5} ki te \frac{5a-3}{5} 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.

$$25a^{2}-35a+12=25\times \frac{\left(5a-4\right)\left(5a-3\right)}{5\times 5}$$

Whakareatia 5 ki te 5.

$$25a^{2}-35a+12=25\times \frac{\left(5a-4\right)\left(5a-3\right)}{25}$$

Whakakorea atu te tauwehe pūnoa nui rawa 25 i roto i te 25 me te 25.

$$25a^{2}-35a+12=\left(5a-4\right)\left(5a-3\right)$$

Aromātai

$$\left(5a-4\right)\left(5a-3\right)$$