Whakaoti i te 660x^2+524x+85

$$660 { x }^{ 2 } +524x+85$$

Tauwehe

$$\left(22x+5\right)\left(30x+17\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 660x^{2}+ax+bx+85. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=524 ab=660\times 85=56100$$

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 56100.

$$1,56100 2,28050 3,18700 4,14025 5,11220 6,9350 10,5610 11,5100 12,4675 15,3740 17,3300 20,2805 22,2550 25,2244 30,1870 33,1700 34,1650 44,1275 50,1122 51,1100 55,1020 60,935 66,850 68,825 75,748 85,660 100,561 102,550 110,510 132,425 150,374 165,340 170,330 187,300 204,275 220,255$$

Tātaihia te tapeke mō ia takirua.

$$1+56100=56101 2+28050=28052 3+18700=18703 4+14025=14029 5+11220=11225 6+9350=9356 10+5610=5620 11+5100=5111 12+4675=4687 15+3740=3755 17+3300=3317 20+2805=2825 22+2550=2572 25+2244=2269 30+1870=1900 33+1700=1733 34+1650=1684 44+1275=1319 50+1122=1172 51+1100=1151 55+1020=1075 60+935=995 66+850=916 68+825=893 75+748=823 85+660=745 100+561=661 102+550=652 110+510=620 132+425=557 150+374=524 165+340=505 170+330=500 187+300=487 204+275=479 220+255=475$$

Ko te otinga te takirua ka hoatu i te tapeke 524.

$$a=150 b=374$$

Tuhia anō te 660x^{2}+524x+85 hei \left(660x^{2}+150x\right)+\left(374x+85\right).

$$\left(660x^{2}+150x\right)+\left(374x+85\right)$$

Tauwehea te 30x i te tuatahi me te 17 i te rōpū tuarua.

$$30x\left(22x+5\right)+17\left(22x+5\right)$$

Whakatauwehea atu te kīanga pātahi 22x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(22x+5\right)\left(30x+17\right)$$

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.

$$660x^{2}+524x+85=0$$

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.

$$x=\frac{-524±\sqrt{524^{2}-4\times 660\times 85}}{2\times 660}$$

Pūrua 524.

$$x=\frac{-524±\sqrt{274576-4\times 660\times 85}}{2\times 660}$$

Whakareatia -4 ki te 660.

$$x=\frac{-524±\sqrt{274576-2640\times 85}}{2\times 660}$$

Whakareatia -2640 ki te 85.

$$x=\frac{-524±\sqrt{274576-224400}}{2\times 660}$$

Tāpiri 274576 ki te -224400.

$$x=\frac{-524±\sqrt{50176}}{2\times 660}$$

Tuhia te pūtakerua o te 50176.

$$x=\frac{-524±224}{2\times 660}$$

Whakareatia 2 ki te 660.

$$x=\frac{-524±224}{1320}$$

Nā, me whakaoti te whārite x=\frac{-524±224}{1320} ina he tāpiri te ±. Tāpiri -524 ki te 224.

$$x=-\frac{300}{1320}$$

Whakahekea te hautanga \frac{-300}{1320} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 60.

$$x=-\frac{5}{22}$$

Nā, me whakaoti te whārite x=\frac{-524±224}{1320} ina he tango te ±. Tango 224 mai i -524.

$$x=-\frac{748}{1320}$$

Whakahekea te hautanga \frac{-748}{1320} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 44.

$$x=-\frac{17}{30}$$

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}{22} mō te x_{1} me te -\frac{17}{30} mō te x_{2}.

$$660x^{2}+524x+85=660\left(x-\left(-\frac{5}{22}\right)\right)\left(x-\left(-\frac{17}{30}\right)\right)$$

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

$$660x^{2}+524x+85=660\left(x+\frac{5}{22}\right)\left(x+\frac{17}{30}\right)$$

Tāpiri \frac{5}{22} 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.

$$660x^{2}+524x+85=660\times \frac{22x+5}{22}\left(x+\frac{17}{30}\right)$$

Tāpiri \frac{17}{30} 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.

$$660x^{2}+524x+85=660\times \frac{22x+5}{22}\times \frac{30x+17}{30}$$

Whakareatia \frac{22x+5}{22} ki te \frac{30x+17}{30} 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.

$$660x^{2}+524x+85=660\times \frac{\left(22x+5\right)\left(30x+17\right)}{22\times 30}$$

Whakareatia 22 ki te 30.

$$660x^{2}+524x+85=660\times \frac{\left(22x+5\right)\left(30x+17\right)}{660}$$

Whakakorea atu te tauwehe pūnoa nui rawa 660 i roto i te 660 me te 660.

$$660x^{2}+524x+85=\left(22x+5\right)\left(30x+17\right)$$

Aromātai

$$\left(22x+5\right)\left(30x+17\right)$$