Whakaoti i te 12x^2+37x+28

$$12 x ^ { 2 } + 37 x + 28$$

Tauwehe

$$\left(3x+4\right)\left(4x+7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 12x^{2}+ax+bx+28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=37 ab=12\times 28=336$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 336.

$$1,336 2,168 3,112 4,84 6,56 7,48 8,42 12,28 14,24 16,21$$

Tātaihia te tapeke mō ia takirua.

$$1+336=337 2+168=170 3+112=115 4+84=88 6+56=62 7+48=55 8+42=50 12+28=40 14+24=38 16+21=37$$

Ko te otinga te takirua ka hoatu i te tapeke 37.

$$a=16 b=21$$

Tuhia anō te 12x^{2}+37x+28 hei \left(12x^{2}+16x\right)+\left(21x+28\right).

$$\left(12x^{2}+16x\right)+\left(21x+28\right)$$

Tauwehea te 4x i te tuatahi me te 7 i te rōpū tuarua.

$$4x\left(3x+4\right)+7\left(3x+4\right)$$

Whakatauwehea atu te kīanga pātahi 3x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+4\right)\left(4x+7\right)$$

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.

$$12x^{2}+37x+28=0$$

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.

$$x=\frac{-37±\sqrt{37^{2}-4\times 12\times 28}}{2\times 12}$$

Pūrua 37.

$$x=\frac{-37±\sqrt{1369-4\times 12\times 28}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-37±\sqrt{1369-48\times 28}}{2\times 12}$$

Whakareatia -48 ki te 28.

$$x=\frac{-37±\sqrt{1369-1344}}{2\times 12}$$

Tāpiri 1369 ki te -1344.

$$x=\frac{-37±\sqrt{25}}{2\times 12}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-37±5}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{-37±5}{24}$$

Nā, me whakaoti te whārite x=\frac{-37±5}{24} ina he tāpiri te ±. Tāpiri -37 ki te 5.

$$x=-\frac{32}{24}$$

Whakahekea te hautanga \frac{-32}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=-\frac{4}{3}$$

Nā, me whakaoti te whārite x=\frac{-37±5}{24} ina he tango te ±. Tango 5 mai i -37.

$$x=-\frac{42}{24}$$

Whakahekea te hautanga \frac{-42}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{7}{4}$$

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

$$12x^{2}+37x+28=12\left(x-\left(-\frac{4}{3}\right)\right)\left(x-\left(-\frac{7}{4}\right)\right)$$

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

$$12x^{2}+37x+28=12\left(x+\frac{4}{3}\right)\left(x+\frac{7}{4}\right)$$

Tāpiri \frac{4}{3} ki te x 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.

$$12x^{2}+37x+28=12\times \frac{3x+4}{3}\left(x+\frac{7}{4}\right)$$

Tāpiri \frac{7}{4} ki te x 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.

$$12x^{2}+37x+28=12\times \frac{3x+4}{3}\times \frac{4x+7}{4}$$

Whakareatia \frac{3x+4}{3} ki te \frac{4x+7}{4} 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.

$$12x^{2}+37x+28=12\times \frac{\left(3x+4\right)\left(4x+7\right)}{3\times 4}$$

Whakareatia 3 ki te 4.

$$12x^{2}+37x+28=12\times \frac{\left(3x+4\right)\left(4x+7\right)}{12}$$

Whakakorea atu te tauwehe pūnoa nui rawa 12 i roto i te 12 me te 12.

$$12x^{2}+37x+28=\left(3x+4\right)\left(4x+7\right)$$

Aromātai

$$\left(3x+4\right)\left(4x+7\right)$$