Whakaoti i te -18a^2-102a+168

$$- 18 a ^ { 2 } - 102 a + 168$$

Tauwehe

$$6\left(-a-7\right)\left(3a-4\right)$$

Ngā Upane Otinga

Tauwehea te 6.

$$6\left(-3a^{2}-17a+28\right)$$

Whakaarohia te -3a^{2}-17a+28. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -3a^{2}+pa+qa+28. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-17 pq=-3\times 28=-84$$

I te mea kua tōraro te pq, he tauaro ngā tohu o p me q. I te mea kua tōraro te p+q, 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 -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -17.

$$p=4 q=-21$$

Tuhia anō te -3a^{2}-17a+28 hei \left(-3a^{2}+4a\right)+\left(-21a+28\right).

$$\left(-3a^{2}+4a\right)+\left(-21a+28\right)$$

Tauwehea te -a i te tuatahi me te -7 i te rōpū tuarua.

$$-a\left(3a-4\right)-7\left(3a-4\right)$$

Whakatauwehea atu te kīanga pātahi 3a-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3a-4\right)\left(-a-7\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$6\left(3a-4\right)\left(-a-7\right)$$

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.

$$-18a^{2}-102a+168=0$$

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.

$$a=\frac{-\left(-102\right)±\sqrt{\left(-102\right)^{2}-4\left(-18\right)\times 168}}{2\left(-18\right)}$$

Pūrua -102.

$$a=\frac{-\left(-102\right)±\sqrt{10404-4\left(-18\right)\times 168}}{2\left(-18\right)}$$

Whakareatia -4 ki te -18.

$$a=\frac{-\left(-102\right)±\sqrt{10404+72\times 168}}{2\left(-18\right)}$$

Whakareatia 72 ki te 168.

$$a=\frac{-\left(-102\right)±\sqrt{10404+12096}}{2\left(-18\right)}$$

Tāpiri 10404 ki te 12096.

$$a=\frac{-\left(-102\right)±\sqrt{22500}}{2\left(-18\right)}$$

Tuhia te pūtakerua o te 22500.

$$a=\frac{-\left(-102\right)±150}{2\left(-18\right)}$$

Ko te tauaro o -102 ko 102.

$$a=\frac{102±150}{2\left(-18\right)}$$

Whakareatia 2 ki te -18.

$$a=\frac{102±150}{-36}$$

Nā, me whakaoti te whārite a=\frac{102±150}{-36} ina he tāpiri te ±. Tāpiri 102 ki te 150.

$$a=\frac{252}{-36}$$

Whakawehe 252 ki te -36.

$$a=-7$$

Nā, me whakaoti te whārite a=\frac{102±150}{-36} ina he tango te ±. Tango 150 mai i 102.

$$a=-\frac{48}{-36}$$

Whakahekea te hautanga \frac{-48}{-36} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$a=\frac{4}{3}$$

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 -7 mō te x_{1} me te \frac{4}{3} mō te x_{2}.

$$-18a^{2}-102a+168=-18\left(a-\left(-7\right)\right)\left(a-\frac{4}{3}\right)$$

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

$$-18a^{2}-102a+168=-18\left(a+7\right)\left(a-\frac{4}{3}\right)$$

Tango \frac{4}{3} 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.

$$-18a^{2}-102a+168=-18\left(a+7\right)\times \frac{-3a+4}{-3}$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te -18 me te 3.

$$-18a^{2}-102a+168=6\left(a+7\right)\left(-3a+4\right)$$

Aromātai

$$168-102a-18a^{2}$$