Whakaoti i te 15x^2-8x-16

$$15 { x }^{ 2 } -8x-16$$

Tauwehe

$$\left(3x-4\right)\left(5x+4\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 15x^{2}+ax+bx-16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=15\left(-16\right)=-240$$

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 -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Tātaihia te tapeke mō ia takirua.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-20 b=12$$

Tuhia anō te 15x^{2}-8x-16 hei \left(15x^{2}-20x\right)+\left(12x-16\right).

$$\left(15x^{2}-20x\right)+\left(12x-16\right)$$

Tauwehea te 5x i te tuatahi me te 4 i te rōpū tuarua.

$$5x\left(3x-4\right)+4\left(3x-4\right)$$

Whakatauwehea atu te kīanga pātahi 3x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-4\right)\left(5x+4\right)$$

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.

$$15x^{2}-8x-16=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15\left(-16\right)}}{2\times 15}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 15\left(-16\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-\left(-8\right)±\sqrt{64-60\left(-16\right)}}{2\times 15}$$

Whakareatia -60 ki te -16.

$$x=\frac{-\left(-8\right)±\sqrt{64+960}}{2\times 15}$$

Tāpiri 64 ki te 960.

$$x=\frac{-\left(-8\right)±\sqrt{1024}}{2\times 15}$$

Tuhia te pūtakerua o te 1024.

$$x=\frac{-\left(-8\right)±32}{2\times 15}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±32}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{8±32}{30}$$

Nā, me whakaoti te whārite x=\frac{8±32}{30} ina he tāpiri te ±. Tāpiri 8 ki te 32.

$$x=\frac{40}{30}$$

Whakahekea te hautanga \frac{40}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{4}{3}$$

Nā, me whakaoti te whārite x=\frac{8±32}{30} ina he tango te ±. Tango 32 mai i 8.

$$x=-\frac{24}{30}$$

Whakahekea te hautanga \frac{-24}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{4}{5}$$

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{4}{3} mō te x_{1} me te -\frac{4}{5} mō te x_{2}.

$$15x^{2}-8x-16=15\left(x-\frac{4}{3}\right)\left(x-\left(-\frac{4}{5}\right)\right)$$

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

$$15x^{2}-8x-16=15\left(x-\frac{4}{3}\right)\left(x+\frac{4}{5}\right)$$

Tango \frac{4}{3} mai i x 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.

$$15x^{2}-8x-16=15\times \frac{3x-4}{3}\left(x+\frac{4}{5}\right)$$

Tāpiri \frac{4}{5} 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.

$$15x^{2}-8x-16=15\times \frac{3x-4}{3}\times \frac{5x+4}{5}$$

Whakareatia \frac{3x-4}{3} ki te \frac{5x+4}{5} 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.

$$15x^{2}-8x-16=15\times \frac{\left(3x-4\right)\left(5x+4\right)}{3\times 5}$$

Whakareatia 3 ki te 5.

$$15x^{2}-8x-16=15\times \frac{\left(3x-4\right)\left(5x+4\right)}{15}$$

Whakakorea atu te tauwehe pūnoa nui rawa 15 i roto i te 15 me te 15.

$$15x^{2}-8x-16=\left(3x-4\right)\left(5x+4\right)$$

Aromātai

$$\left(3x-4\right)\left(5x+4\right)$$