Whakaoti i te 24w^2-23w-630

$$24 w ^ { 2 } - 23 w - 630$$

Tauwehe

$$\left(8w-45\right)\left(3w+14\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 24w^{2}+aw+bw-630. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-23 ab=24\left(-630\right)=-15120$$

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 -15120.

$$1,-15120 2,-7560 3,-5040 4,-3780 5,-3024 6,-2520 7,-2160 8,-1890 9,-1680 10,-1512 12,-1260 14,-1080 15,-1008 16,-945 18,-840 20,-756 21,-720 24,-630 27,-560 28,-540 30,-504 35,-432 36,-420 40,-378 42,-360 45,-336 48,-315 54,-280 56,-270 60,-252 63,-240 70,-216 72,-210 80,-189 84,-180 90,-168 105,-144 108,-140 112,-135 120,-126$$

Tātaihia te tapeke mō ia takirua.

$$1-15120=-15119 2-7560=-7558 3-5040=-5037 4-3780=-3776 5-3024=-3019 6-2520=-2514 7-2160=-2153 8-1890=-1882 9-1680=-1671 10-1512=-1502 12-1260=-1248 14-1080=-1066 15-1008=-993 16-945=-929 18-840=-822 20-756=-736 21-720=-699 24-630=-606 27-560=-533 28-540=-512 30-504=-474 35-432=-397 36-420=-384 40-378=-338 42-360=-318 45-336=-291 48-315=-267 54-280=-226 56-270=-214 60-252=-192 63-240=-177 70-216=-146 72-210=-138 80-189=-109 84-180=-96 90-168=-78 105-144=-39 108-140=-32 112-135=-23 120-126=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -23.

$$a=-135 b=112$$

Tuhia anō te 24w^{2}-23w-630 hei \left(24w^{2}-135w\right)+\left(112w-630\right).

$$\left(24w^{2}-135w\right)+\left(112w-630\right)$$

Tauwehea te 3w i te tuatahi me te 14 i te rōpū tuarua.

$$3w\left(8w-45\right)+14\left(8w-45\right)$$

Whakatauwehea atu te kīanga pātahi 8w-45 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(8w-45\right)\left(3w+14\right)$$

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.

$$24w^{2}-23w-630=0$$

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.

$$w=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 24\left(-630\right)}}{2\times 24}$$

Pūrua -23.

$$w=\frac{-\left(-23\right)±\sqrt{529-4\times 24\left(-630\right)}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$w=\frac{-\left(-23\right)±\sqrt{529-96\left(-630\right)}}{2\times 24}$$

Whakareatia -96 ki te -630.

$$w=\frac{-\left(-23\right)±\sqrt{529+60480}}{2\times 24}$$

Tāpiri 529 ki te 60480.

$$w=\frac{-\left(-23\right)±\sqrt{61009}}{2\times 24}$$

Tuhia te pūtakerua o te 61009.

$$w=\frac{-\left(-23\right)±247}{2\times 24}$$

Ko te tauaro o -23 ko 23.

$$w=\frac{23±247}{2\times 24}$$

Whakareatia 2 ki te 24.

$$w=\frac{23±247}{48}$$

Nā, me whakaoti te whārite w=\frac{23±247}{48} ina he tāpiri te ±. Tāpiri 23 ki te 247.

$$w=\frac{270}{48}$$

Whakahekea te hautanga \frac{270}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$w=\frac{45}{8}$$

Nā, me whakaoti te whārite w=\frac{23±247}{48} ina he tango te ±. Tango 247 mai i 23.

$$w=-\frac{224}{48}$$

Whakahekea te hautanga \frac{-224}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$w=-\frac{14}{3}$$

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{45}{8} mō te x_{1} me te -\frac{14}{3} mō te x_{2}.

$$24w^{2}-23w-630=24\left(w-\frac{45}{8}\right)\left(w-\left(-\frac{14}{3}\right)\right)$$

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

$$24w^{2}-23w-630=24\left(w-\frac{45}{8}\right)\left(w+\frac{14}{3}\right)$$

Tango \frac{45}{8} mai i w 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.

$$24w^{2}-23w-630=24\times \frac{8w-45}{8}\left(w+\frac{14}{3}\right)$$

Tāpiri \frac{14}{3} ki te w 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.

$$24w^{2}-23w-630=24\times \frac{8w-45}{8}\times \frac{3w+14}{3}$$

Whakareatia \frac{8w-45}{8} ki te \frac{3w+14}{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.

$$24w^{2}-23w-630=24\times \frac{\left(8w-45\right)\left(3w+14\right)}{8\times 3}$$

Whakareatia 8 ki te 3.

$$24w^{2}-23w-630=24\times \frac{\left(8w-45\right)\left(3w+14\right)}{24}$$

Whakakorea atu te tauwehe pūnoa nui rawa 24 i roto i te 24 me te 24.

$$24w^{2}-23w-630=\left(8w-45\right)\left(3w+14\right)$$

Aromātai

$$\left(8w-45\right)\left(3w+14\right)$$