Whakaoti i te 12x^2+23x-24

$$12 x ^ { 2 } + 23 x - 24$$

Tauwehe

$$\left(4x-3\right)\left(3x+8\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 12x^{2}+ax+bx-24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=23 ab=12\left(-24\right)=-288$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -288.

$$-1,288 -2,144 -3,96 -4,72 -6,48 -8,36 -9,32 -12,24 -16,18$$

Tātaihia te tapeke mō ia takirua.

$$-1+288=287 -2+144=142 -3+96=93 -4+72=68 -6+48=42 -8+36=28 -9+32=23 -12+24=12 -16+18=2$$

Ko te otinga te takirua ka hoatu i te tapeke 23.

$$a=-9 b=32$$

Tuhia anō te 12x^{2}+23x-24 hei \left(12x^{2}-9x\right)+\left(32x-24\right).

$$\left(12x^{2}-9x\right)+\left(32x-24\right)$$

Tauwehea te 3x i te tuatahi me te 8 i te rōpū tuarua.

$$3x\left(4x-3\right)+8\left(4x-3\right)$$

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

$$\left(4x-3\right)\left(3x+8\right)$$

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.

$$12x^{2}+23x-24=0$$

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.

$$x=\frac{-23±\sqrt{23^{2}-4\times 12\left(-24\right)}}{2\times 12}$$

Pūrua 23.

$$x=\frac{-23±\sqrt{529-4\times 12\left(-24\right)}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-23±\sqrt{529-48\left(-24\right)}}{2\times 12}$$

Whakareatia -48 ki te -24.

$$x=\frac{-23±\sqrt{529+1152}}{2\times 12}$$

Tāpiri 529 ki te 1152.

$$x=\frac{-23±\sqrt{1681}}{2\times 12}$$

Tuhia te pūtakerua o te 1681.

$$x=\frac{-23±41}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{-23±41}{24}$$

Nā, me whakaoti te whārite x=\frac{-23±41}{24} ina he tāpiri te ±. Tāpiri -23 ki te 41.

$$x=\frac{18}{24}$$

Whakahekea te hautanga \frac{18}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Nā, me whakaoti te whārite x=\frac{-23±41}{24} ina he tango te ±. Tango 41 mai i -23.

$$x=-\frac{64}{24}$$

Whakahekea te hautanga \frac{-64}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=-\frac{8}{3}$$

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 \frac{3}{4} mō te x_{1} me te -\frac{8}{3} mō te x_{2}.

$$12x^{2}+23x-24=12\left(x-\frac{3}{4}\right)\left(x-\left(-\frac{8}{3}\right)\right)$$

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

$$12x^{2}+23x-24=12\left(x-\frac{3}{4}\right)\left(x+\frac{8}{3}\right)$$

Tango \frac{3}{4} mai i x 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.

$$12x^{2}+23x-24=12\times \frac{4x-3}{4}\left(x+\frac{8}{3}\right)$$

Tāpiri \frac{8}{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.

$$12x^{2}+23x-24=12\times \frac{4x-3}{4}\times \frac{3x+8}{3}$$

Whakareatia \frac{4x-3}{4} ki te \frac{3x+8}{3} 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.

$$12x^{2}+23x-24=12\times \frac{\left(4x-3\right)\left(3x+8\right)}{4\times 3}$$

Whakareatia 4 ki te 3.

$$12x^{2}+23x-24=12\times \frac{\left(4x-3\right)\left(3x+8\right)}{12}$$

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

$$12x^{2}+23x-24=\left(4x-3\right)\left(3x+8\right)$$

Aromātai

$$\left(4x-3\right)\left(3x+8\right)$$