Whakaoti i te 77r^2+45r-18

$$77 r ^ { 2 } + 45 r - 18$$

Tauwehe

$$\left(11r-3\right)\left(7r+6\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 77r^{2}+ar+br-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=45 ab=77\left(-18\right)=-1386$$

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 -1386.

$$-1,1386 -2,693 -3,462 -6,231 -7,198 -9,154 -11,126 -14,99 -18,77 -21,66 -22,63 -33,42$$

Tātaihia te tapeke mō ia takirua.

$$-1+1386=1385 -2+693=691 -3+462=459 -6+231=225 -7+198=191 -9+154=145 -11+126=115 -14+99=85 -18+77=59 -21+66=45 -22+63=41 -33+42=9$$

Ko te otinga te takirua ka hoatu i te tapeke 45.

$$a=-21 b=66$$

Tuhia anō te 77r^{2}+45r-18 hei \left(77r^{2}-21r\right)+\left(66r-18\right).

$$\left(77r^{2}-21r\right)+\left(66r-18\right)$$

Tauwehea te 7r i te tuatahi me te 6 i te rōpū tuarua.

$$7r\left(11r-3\right)+6\left(11r-3\right)$$

Whakatauwehea atu te kīanga pātahi 11r-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(11r-3\right)\left(7r+6\right)$$

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.

$$77r^{2}+45r-18=0$$

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.

$$r=\frac{-45±\sqrt{45^{2}-4\times 77\left(-18\right)}}{2\times 77}$$

Pūrua 45.

$$r=\frac{-45±\sqrt{2025-4\times 77\left(-18\right)}}{2\times 77}$$

Whakareatia -4 ki te 77.

$$r=\frac{-45±\sqrt{2025-308\left(-18\right)}}{2\times 77}$$

Whakareatia -308 ki te -18.

$$r=\frac{-45±\sqrt{2025+5544}}{2\times 77}$$

Tāpiri 2025 ki te 5544.

$$r=\frac{-45±\sqrt{7569}}{2\times 77}$$

Tuhia te pūtakerua o te 7569.

$$r=\frac{-45±87}{2\times 77}$$

Whakareatia 2 ki te 77.

$$r=\frac{-45±87}{154}$$

Nā, me whakaoti te whārite r=\frac{-45±87}{154} ina he tāpiri te ±. Tāpiri -45 ki te 87.

$$r=\frac{42}{154}$$

Whakahekea te hautanga \frac{42}{154} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$r=\frac{3}{11}$$

Nā, me whakaoti te whārite r=\frac{-45±87}{154} ina he tango te ±. Tango 87 mai i -45.

$$r=-\frac{132}{154}$$

Whakahekea te hautanga \frac{-132}{154} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 22.

$$r=-\frac{6}{7}$$

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{3}{11} mō te x_{1} me te -\frac{6}{7} mō te x_{2}.

$$77r^{2}+45r-18=77\left(r-\frac{3}{11}\right)\left(r-\left(-\frac{6}{7}\right)\right)$$

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

$$77r^{2}+45r-18=77\left(r-\frac{3}{11}\right)\left(r+\frac{6}{7}\right)$$

Tango \frac{3}{11} mai i r 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.

$$77r^{2}+45r-18=77\times \frac{11r-3}{11}\left(r+\frac{6}{7}\right)$$

Tāpiri \frac{6}{7} ki te r 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.

$$77r^{2}+45r-18=77\times \frac{11r-3}{11}\times \frac{7r+6}{7}$$

Whakareatia \frac{11r-3}{11} ki te \frac{7r+6}{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.

$$77r^{2}+45r-18=77\times \frac{\left(11r-3\right)\left(7r+6\right)}{11\times 7}$$

Whakareatia 11 ki te 7.

$$77r^{2}+45r-18=77\times \frac{\left(11r-3\right)\left(7r+6\right)}{77}$$

Whakakorea atu te tauwehe pūnoa nui rawa 77 i roto i te 77 me te 77.

$$77r^{2}+45r-18=\left(11r-3\right)\left(7r+6\right)$$

Aromātai

$$\left(11r-3\right)\left(7r+6\right)$$