Whakaoti i te 6x^2+13x-28

$$6 x ^ { 2 } + 13 x - 28$$

Tauwehe

$$\left(3x-4\right)\left(2x+7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6x^{2}+ax+bx-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=6\left(-28\right)=-168$$

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 -168.

$$-1,168 -2,84 -3,56 -4,42 -6,28 -7,24 -8,21 -12,14$$

Tātaihia te tapeke mō ia takirua.

$$-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=-8 b=21$$

Tuhia anō te 6x^{2}+13x-28 hei \left(6x^{2}-8x\right)+\left(21x-28\right).

$$\left(6x^{2}-8x\right)+\left(21x-28\right)$$

Tauwehea te 2x i te tuatahi me te 7 i te rōpū tuarua.

$$2x\left(3x-4\right)+7\left(3x-4\right)$$

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

$$\left(3x-4\right)\left(2x+7\right)$$

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.

$$6x^{2}+13x-28=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\times 6\left(-28\right)}}{2\times 6}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\times 6\left(-28\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-13±\sqrt{169-24\left(-28\right)}}{2\times 6}$$

Whakareatia -24 ki te -28.

$$x=\frac{-13±\sqrt{169+672}}{2\times 6}$$

Tāpiri 169 ki te 672.

$$x=\frac{-13±\sqrt{841}}{2\times 6}$$

Tuhia te pūtakerua o te 841.

$$x=\frac{-13±29}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-13±29}{12}$$

Nā, me whakaoti te whārite x=\frac{-13±29}{12} ina he tāpiri te ±. Tāpiri -13 ki te 29.

$$x=\frac{16}{12}$$

Whakahekea te hautanga \frac{16}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite x=\frac{-13±29}{12} ina he tango te ±. Tango 29 mai i -13.

$$x=-\frac{42}{12}$$

Whakahekea te hautanga \frac{-42}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{7}{2}$$

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}{2} mō te x_{2}.

$$6x^{2}+13x-28=6\left(x-\frac{4}{3}\right)\left(x-\left(-\frac{7}{2}\right)\right)$$

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

$$6x^{2}+13x-28=6\left(x-\frac{4}{3}\right)\left(x+\frac{7}{2}\right)$$

Tango \frac{4}{3} 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.

$$6x^{2}+13x-28=6\times \frac{3x-4}{3}\left(x+\frac{7}{2}\right)$$

Tāpiri \frac{7}{2} 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.

$$6x^{2}+13x-28=6\times \frac{3x-4}{3}\times \frac{2x+7}{2}$$

Whakareatia \frac{3x-4}{3} ki te \frac{2x+7}{2} 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.

$$6x^{2}+13x-28=6\times \frac{\left(3x-4\right)\left(2x+7\right)}{3\times 2}$$

Whakareatia 3 ki te 2.

$$6x^{2}+13x-28=6\times \frac{\left(3x-4\right)\left(2x+7\right)}{6}$$

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

$$6x^{2}+13x-28=\left(3x-4\right)\left(2x+7\right)$$

Aromātai

$$\left(3x-4\right)\left(2x+7\right)$$