Whakaoti i te 30s^2-19s-63

$$30 s ^ { 2 } - 19 s - 63$$

Tauwehe

$$\left(5s-9\right)\left(6s+7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 30s^{2}+as+bs-63. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-19 ab=30\left(-63\right)=-1890$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -1890.

$$1,-1890 2,-945 3,-630 5,-378 6,-315 7,-270 9,-210 10,-189 14,-135 15,-126 18,-105 21,-90 27,-70 30,-63 35,-54 42,-45$$

Tātaihia te tapeke mō ia takirua.

$$1-1890=-1889 2-945=-943 3-630=-627 5-378=-373 6-315=-309 7-270=-263 9-210=-201 10-189=-179 14-135=-121 15-126=-111 18-105=-87 21-90=-69 27-70=-43 30-63=-33 35-54=-19 42-45=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -19.

$$a=-54 b=35$$

Tuhia anō te 30s^{2}-19s-63 hei \left(30s^{2}-54s\right)+\left(35s-63\right).

$$\left(30s^{2}-54s\right)+\left(35s-63\right)$$

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

$$6s\left(5s-9\right)+7\left(5s-9\right)$$

Whakatauwehea atu te kīanga pātahi 5s-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5s-9\right)\left(6s+7\right)$$

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.

$$30s^{2}-19s-63=0$$

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.

$$s=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 30\left(-63\right)}}{2\times 30}$$

Pūrua -19.

$$s=\frac{-\left(-19\right)±\sqrt{361-4\times 30\left(-63\right)}}{2\times 30}$$

Whakareatia -4 ki te 30.

$$s=\frac{-\left(-19\right)±\sqrt{361-120\left(-63\right)}}{2\times 30}$$

Whakareatia -120 ki te -63.

$$s=\frac{-\left(-19\right)±\sqrt{361+7560}}{2\times 30}$$

Tāpiri 361 ki te 7560.

$$s=\frac{-\left(-19\right)±\sqrt{7921}}{2\times 30}$$

Tuhia te pūtakerua o te 7921.

$$s=\frac{-\left(-19\right)±89}{2\times 30}$$

Ko te tauaro o -19 ko 19.

$$s=\frac{19±89}{2\times 30}$$

Whakareatia 2 ki te 30.

$$s=\frac{19±89}{60}$$

Nā, me whakaoti te whārite s=\frac{19±89}{60} ina he tāpiri te ±. Tāpiri 19 ki te 89.

$$s=\frac{108}{60}$$

Whakahekea te hautanga \frac{108}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$s=\frac{9}{5}$$

Nā, me whakaoti te whārite s=\frac{19±89}{60} ina he tango te ±. Tango 89 mai i 19.

$$s=-\frac{70}{60}$$

Whakahekea te hautanga \frac{-70}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

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

$$30s^{2}-19s-63=30\left(s-\frac{9}{5}\right)\left(s-\left(-\frac{7}{6}\right)\right)$$

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

$$30s^{2}-19s-63=30\left(s-\frac{9}{5}\right)\left(s+\frac{7}{6}\right)$$

Tango \frac{9}{5} mai i s 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.

$$30s^{2}-19s-63=30\times \frac{5s-9}{5}\left(s+\frac{7}{6}\right)$$

Tāpiri \frac{7}{6} ki te s 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.

$$30s^{2}-19s-63=30\times \frac{5s-9}{5}\times \frac{6s+7}{6}$$

Whakareatia \frac{5s-9}{5} ki te \frac{6s+7}{6} 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.

$$30s^{2}-19s-63=30\times \frac{\left(5s-9\right)\left(6s+7\right)}{5\times 6}$$

Whakareatia 5 ki te 6.

$$30s^{2}-19s-63=30\times \frac{\left(5s-9\right)\left(6s+7\right)}{30}$$

Whakakorea atu te tauwehe pūnoa nui rawa 30 i roto i te 30 me te 30.

$$30s^{2}-19s-63=\left(5s-9\right)\left(6s+7\right)$$

Aromātai

$$\left(5s-9\right)\left(6s+7\right)$$