Whakaoti i te 24x^2+10x-21

$$24 { x }^{ 2 } +10x-21$$

Tauwehe

$$\left(4x-3\right)\left(6x+7\right)$$

Ngā Upane Otinga

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

$$a+b=10 ab=24\left(-21\right)=-504$$

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 -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.

$$-1+504=503 -2+252=250 -3+168=165 -4+126=122 -6+84=78 -7+72=65 -8+63=55 -9+56=47 -12+42=30 -14+36=22 -18+28=10 -21+24=3$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-18 b=28$$

Tuhia anō te 24x^{2}+10x-21 hei \left(24x^{2}-18x\right)+\left(28x-21\right).

$$\left(24x^{2}-18x\right)+\left(28x-21\right)$$

Tauwehea te 6x i te tuatahi me te 7 i te rōpū tuarua.

$$6x\left(4x-3\right)+7\left(4x-3\right)$$

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

$$\left(4x-3\right)\left(6x+7\right)$$

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.

$$24x^{2}+10x-21=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\times 24\left(-21\right)}}{2\times 24}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 24\left(-21\right)}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$x=\frac{-10±\sqrt{100-96\left(-21\right)}}{2\times 24}$$

Whakareatia -96 ki te -21.

$$x=\frac{-10±\sqrt{100+2016}}{2\times 24}$$

Tāpiri 100 ki te 2016.

$$x=\frac{-10±\sqrt{2116}}{2\times 24}$$

Tuhia te pūtakerua o te 2116.

$$x=\frac{-10±46}{2\times 24}$$

Whakareatia 2 ki te 24.

$$x=\frac{-10±46}{48}$$

Nā, me whakaoti te whārite x=\frac{-10±46}{48} ina he tāpiri te ±. Tāpiri -10 ki te 46.

$$x=\frac{36}{48}$$

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

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

Nā, me whakaoti te whārite x=\frac{-10±46}{48} ina he tango te ±. Tango 46 mai i -10.

$$x=-\frac{56}{48}$$

Whakahekea te hautanga \frac{-56}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=-\frac{7}{6}$$

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{7}{6} mō te x_{2}.

$$24x^{2}+10x-21=24\left(x-\frac{3}{4}\right)\left(x-\left(-\frac{7}{6}\right)\right)$$

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

$$24x^{2}+10x-21=24\left(x-\frac{3}{4}\right)\left(x+\frac{7}{6}\right)$$

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.

$$24x^{2}+10x-21=24\times \frac{4x-3}{4}\left(x+\frac{7}{6}\right)$$

Tāpiri \frac{7}{6} 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.

$$24x^{2}+10x-21=24\times \frac{4x-3}{4}\times \frac{6x+7}{6}$$

Whakareatia \frac{4x-3}{4} ki te \frac{6x+7}{6} 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.

$$24x^{2}+10x-21=24\times \frac{\left(4x-3\right)\left(6x+7\right)}{4\times 6}$$

Whakareatia 4 ki te 6.

$$24x^{2}+10x-21=24\times \frac{\left(4x-3\right)\left(6x+7\right)}{24}$$

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

$$24x^{2}+10x-21=\left(4x-3\right)\left(6x+7\right)$$

Aromātai

$$\left(4x-3\right)\left(6x+7\right)$$