Whakaoti i te 21x^2+10x-16

$$21 x ^ { 2 } + 10 x - 16$$

Tauwehe

$$\left(3x-2\right)\left(7x+8\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 21x^{2}+ax+bx-16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=21\left(-16\right)=-336$$

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 -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.

$$-1+336=335 -2+168=166 -3+112=109 -4+84=80 -6+56=50 -7+48=41 -8+42=34 -12+28=16 -14+24=10 -16+21=5$$

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

$$a=-14 b=24$$

Tuhia anō te 21x^{2}+10x-16 hei \left(21x^{2}-14x\right)+\left(24x-16\right).

$$\left(21x^{2}-14x\right)+\left(24x-16\right)$$

Tauwehea te 7x i te tuatahi me te 8 i te rōpū tuarua.

$$7x\left(3x-2\right)+8\left(3x-2\right)$$

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

$$\left(3x-2\right)\left(7x+8\right)$$

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.

$$21x^{2}+10x-16=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\times 21\left(-16\right)}}{2\times 21}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 21\left(-16\right)}}{2\times 21}$$

Whakareatia -4 ki te 21.

$$x=\frac{-10±\sqrt{100-84\left(-16\right)}}{2\times 21}$$

Whakareatia -84 ki te -16.

$$x=\frac{-10±\sqrt{100+1344}}{2\times 21}$$

Tāpiri 100 ki te 1344.

$$x=\frac{-10±\sqrt{1444}}{2\times 21}$$

Tuhia te pūtakerua o te 1444.

$$x=\frac{-10±38}{2\times 21}$$

Whakareatia 2 ki te 21.

$$x=\frac{-10±38}{42}$$

Nā, me whakaoti te whārite x=\frac{-10±38}{42} ina he tāpiri te ±. Tāpiri -10 ki te 38.

$$x=\frac{28}{42}$$

Whakahekea te hautanga \frac{28}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

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

Nā, me whakaoti te whārite x=\frac{-10±38}{42} ina he tango te ±. Tango 38 mai i -10.

$$x=-\frac{48}{42}$$

Whakahekea te hautanga \frac{-48}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{8}{7}$$

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{2}{3} mō te x_{1} me te -\frac{8}{7} mō te x_{2}.

$$21x^{2}+10x-16=21\left(x-\frac{2}{3}\right)\left(x-\left(-\frac{8}{7}\right)\right)$$

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

$$21x^{2}+10x-16=21\left(x-\frac{2}{3}\right)\left(x+\frac{8}{7}\right)$$

Tango \frac{2}{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.

$$21x^{2}+10x-16=21\times \frac{3x-2}{3}\left(x+\frac{8}{7}\right)$$

Tāpiri \frac{8}{7} 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.

$$21x^{2}+10x-16=21\times \frac{3x-2}{3}\times \frac{7x+8}{7}$$

Whakareatia \frac{3x-2}{3} ki te \frac{7x+8}{7} 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.

$$21x^{2}+10x-16=21\times \frac{\left(3x-2\right)\left(7x+8\right)}{3\times 7}$$

Whakareatia 3 ki te 7.

$$21x^{2}+10x-16=21\times \frac{\left(3x-2\right)\left(7x+8\right)}{21}$$

Whakakorea atu te tauwehe pūnoa nui rawa 21 i roto i te 21 me te 21.

$$21x^{2}+10x-16=\left(3x-2\right)\left(7x+8\right)$$

Aromātai

$$\left(3x-2\right)\left(7x+8\right)$$