Whakaoti i te 84x^2+70x-294

$$84 { x }^{ 2 } +70x-294$$

Tauwehe

$$14\left(2x-3\right)\left(3x+7\right)$$

Ngā Upane Otinga

Tauwehea te 14.

$$14\left(6x^{2}+5x-21\right)$$

Whakaarohia te 6x^{2}+5x-21. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6x^{2}+ax+bx-21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=6\left(-21\right)=-126$$

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 -126.

$$-1,126 -2,63 -3,42 -6,21 -7,18 -9,14$$

Tātaihia te tapeke mō ia takirua.

$$-1+126=125 -2+63=61 -3+42=39 -6+21=15 -7+18=11 -9+14=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-9 b=14$$

Tuhia anō te 6x^{2}+5x-21 hei \left(6x^{2}-9x\right)+\left(14x-21\right).

$$\left(6x^{2}-9x\right)+\left(14x-21\right)$$

Tauwehea te 3x i te tuatahi me te 7 i te rōpū tuarua.

$$3x\left(2x-3\right)+7\left(2x-3\right)$$

Whakatauwehea atu te kīanga pātahi 2x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-3\right)\left(3x+7\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$14\left(2x-3\right)\left(3x+7\right)$$

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.

$$84x^{2}+70x-294=0$$

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.

$$x=\frac{-70±\sqrt{70^{2}-4\times 84\left(-294\right)}}{2\times 84}$$

Pūrua 70.

$$x=\frac{-70±\sqrt{4900-4\times 84\left(-294\right)}}{2\times 84}$$

Whakareatia -4 ki te 84.

$$x=\frac{-70±\sqrt{4900-336\left(-294\right)}}{2\times 84}$$

Whakareatia -336 ki te -294.

$$x=\frac{-70±\sqrt{4900+98784}}{2\times 84}$$

Tāpiri 4900 ki te 98784.

$$x=\frac{-70±\sqrt{103684}}{2\times 84}$$

Tuhia te pūtakerua o te 103684.

$$x=\frac{-70±322}{2\times 84}$$

Whakareatia 2 ki te 84.

$$x=\frac{-70±322}{168}$$

Nā, me whakaoti te whārite x=\frac{-70±322}{168} ina he tāpiri te ±. Tāpiri -70 ki te 322.

$$x=\frac{252}{168}$$

Whakahekea te hautanga \frac{252}{168} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 84.

$$x=\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{-70±322}{168} ina he tango te ±. Tango 322 mai i -70.

$$x=-\frac{392}{168}$$

Whakahekea te hautanga \frac{-392}{168} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 56.

$$x=-\frac{7}{3}$$

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

$$84x^{2}+70x-294=84\left(x-\frac{3}{2}\right)\left(x-\left(-\frac{7}{3}\right)\right)$$

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

$$84x^{2}+70x-294=84\left(x-\frac{3}{2}\right)\left(x+\frac{7}{3}\right)$$

Tango \frac{3}{2} 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.

$$84x^{2}+70x-294=84\times \frac{2x-3}{2}\left(x+\frac{7}{3}\right)$$

Tāpiri \frac{7}{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.

$$84x^{2}+70x-294=84\times \frac{2x-3}{2}\times \frac{3x+7}{3}$$

Whakareatia \frac{2x-3}{2} ki te \frac{3x+7}{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.

$$84x^{2}+70x-294=84\times \frac{\left(2x-3\right)\left(3x+7\right)}{2\times 3}$$

Whakareatia 2 ki te 3.

$$84x^{2}+70x-294=84\times \frac{\left(2x-3\right)\left(3x+7\right)}{6}$$

Whakakorea atu te tauwehe pūnoa nui rawa 6 i roto i te 84 me te 6.

$$84x^{2}+70x-294=14\left(2x-3\right)\left(3x+7\right)$$

Aromātai

$$84x^{2}+70x-294$$