Whakaoti i te 21x^2+55x+36

$$21 { x }^{ 2 } +55x+36$$

Tauwehe

$$\left(3x+4\right)\left(7x+9\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 21x^{2}+ax+bx+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=55 ab=21\times 36=756$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 756.

$$1,756 2,378 3,252 4,189 6,126 7,108 9,84 12,63 14,54 18,42 21,36 27,28$$

Tātaihia te tapeke mō ia takirua.

$$1+756=757 2+378=380 3+252=255 4+189=193 6+126=132 7+108=115 9+84=93 12+63=75 14+54=68 18+42=60 21+36=57 27+28=55$$

Ko te otinga te takirua ka hoatu i te tapeke 55.

$$a=27 b=28$$

Tuhia anō te 21x^{2}+55x+36 hei \left(21x^{2}+27x\right)+\left(28x+36\right).

$$\left(21x^{2}+27x\right)+\left(28x+36\right)$$

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

$$3x\left(7x+9\right)+4\left(7x+9\right)$$

Whakatauwehea atu te kīanga pātahi 7x+9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x+9\right)\left(3x+4\right)$$

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.

$$21x^{2}+55x+36=0$$

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.

$$x=\frac{-55±\sqrt{55^{2}-4\times 21\times 36}}{2\times 21}$$

Pūrua 55.

$$x=\frac{-55±\sqrt{3025-4\times 21\times 36}}{2\times 21}$$

Whakareatia -4 ki te 21.

$$x=\frac{-55±\sqrt{3025-84\times 36}}{2\times 21}$$

Whakareatia -84 ki te 36.

$$x=\frac{-55±\sqrt{3025-3024}}{2\times 21}$$

Tāpiri 3025 ki te -3024.

$$x=\frac{-55±\sqrt{1}}{2\times 21}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-55±1}{2\times 21}$$

Whakareatia 2 ki te 21.

$$x=\frac{-55±1}{42}$$

Nā, me whakaoti te whārite x=\frac{-55±1}{42} ina he tāpiri te ±. Tāpiri -55 ki te 1.

$$x=-\frac{54}{42}$$

Whakahekea te hautanga \frac{-54}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{9}{7}$$

Nā, me whakaoti te whārite x=\frac{-55±1}{42} ina he tango te ±. Tango 1 mai i -55.

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

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{9}{7} mō te x_{1} me te -\frac{4}{3} mō te x_{2}.

$$21x^{2}+55x+36=21\left(x-\left(-\frac{9}{7}\right)\right)\left(x-\left(-\frac{4}{3}\right)\right)$$

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

$$21x^{2}+55x+36=21\left(x+\frac{9}{7}\right)\left(x+\frac{4}{3}\right)$$

Tāpiri \frac{9}{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.

$$21x^{2}+55x+36=21\times \frac{7x+9}{7}\left(x+\frac{4}{3}\right)$$

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.

$$21x^{2}+55x+36=21\times \frac{7x+9}{7}\times \frac{3x+4}{3}$$

Whakareatia \frac{7x+9}{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.

$$21x^{2}+55x+36=21\times \frac{\left(7x+9\right)\left(3x+4\right)}{7\times 3}$$

Whakareatia 7 ki te 3.

$$21x^{2}+55x+36=21\times \frac{\left(7x+9\right)\left(3x+4\right)}{21}$$

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

$$21x^{2}+55x+36=\left(7x+9\right)\left(3x+4\right)$$

Aromātai

$$\left(3x+4\right)\left(7x+9\right)$$