Whakaoti i te 21x^2+13x-20

$$21 x ^ { 2 } + 13 x - 20$$

Tauwehe

$$\left(7x-5\right)\left(3x+4\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 21x^{2}+ax+bx-20. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=21\left(-20\right)=-420$$

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 -420.

$$-1,420 -2,210 -3,140 -4,105 -5,84 -6,70 -7,60 -10,42 -12,35 -14,30 -15,28 -20,21$$

Tātaihia te tapeke mō ia takirua.

$$-1+420=419 -2+210=208 -3+140=137 -4+105=101 -5+84=79 -6+70=64 -7+60=53 -10+42=32 -12+35=23 -14+30=16 -15+28=13 -20+21=1$$

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

$$a=-15 b=28$$

Tuhia anō te 21x^{2}+13x-20 hei \left(21x^{2}-15x\right)+\left(28x-20\right).

$$\left(21x^{2}-15x\right)+\left(28x-20\right)$$

Tauwehea te 3x i te tuatahi me te 4 i te rōpū tuarua.

$$3x\left(7x-5\right)+4\left(7x-5\right)$$

Whakatauwehea atu te kīanga pātahi 7x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-5\right)\left(3x+4\right)$$

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.

$$21x^{2}+13x-20=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\times 21\left(-20\right)}}{2\times 21}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\times 21\left(-20\right)}}{2\times 21}$$

Whakareatia -4 ki te 21.

$$x=\frac{-13±\sqrt{169-84\left(-20\right)}}{2\times 21}$$

Whakareatia -84 ki te -20.

$$x=\frac{-13±\sqrt{169+1680}}{2\times 21}$$

Tāpiri 169 ki te 1680.

$$x=\frac{-13±\sqrt{1849}}{2\times 21}$$

Tuhia te pūtakerua o te 1849.

$$x=\frac{-13±43}{2\times 21}$$

Whakareatia 2 ki te 21.

$$x=\frac{-13±43}{42}$$

Nā, me whakaoti te whārite x=\frac{-13±43}{42} ina he tāpiri te ±. Tāpiri -13 ki te 43.

$$x=\frac{30}{42}$$

Whakahekea te hautanga \frac{30}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{5}{7}$$

Nā, me whakaoti te whārite x=\frac{-13±43}{42} ina he tango te ±. Tango 43 mai i -13.

$$x=-\frac{56}{42}$$

Whakahekea te hautanga \frac{-56}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

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

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{5}{7} mō te x_{1} me te -\frac{4}{3} mō te x_{2}.

$$21x^{2}+13x-20=21\left(x-\frac{5}{7}\right)\left(x-\left(-\frac{4}{3}\right)\right)$$

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

$$21x^{2}+13x-20=21\left(x-\frac{5}{7}\right)\left(x+\frac{4}{3}\right)$$

Tango \frac{5}{7} 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.

$$21x^{2}+13x-20=21\times \frac{7x-5}{7}\left(x+\frac{4}{3}\right)$$

Tāpiri \frac{4}{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.

$$21x^{2}+13x-20=21\times \frac{7x-5}{7}\times \frac{3x+4}{3}$$

Whakareatia \frac{7x-5}{7} ki te \frac{3x+4}{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.

$$21x^{2}+13x-20=21\times \frac{\left(7x-5\right)\left(3x+4\right)}{7\times 3}$$

Whakareatia 7 ki te 3.

$$21x^{2}+13x-20=21\times \frac{\left(7x-5\right)\left(3x+4\right)}{21}$$

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

$$21x^{2}+13x-20=\left(7x-5\right)\left(3x+4\right)$$

Aromātai

$$\left(7x-5\right)\left(3x+4\right)$$