Whakaoti i te 15x^2-25x-60

$$15 x ^ { 2 } - 25 x - 60$$

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 5.

$$5\left(3x^{2}-5x-12\right)$$

Whakaarohia te 3x^{2}-5x-12. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 3x^{2}+ax+bx-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=3\left(-12\right)=-36$$

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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-9 b=4$$

Tuhia anō te 3x^{2}-5x-12 hei \left(3x^{2}-9x\right)+\left(4x-12\right).

$$\left(3x^{2}-9x\right)+\left(4x-12\right)$$

Tauwehea te 3x i te tuatahi me te 4 i te rōpū tuarua.

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

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

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.

$$15x^{2}-25x-60=0$$

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.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 15\left(-60\right)}}{2\times 15}$$

Pūrua -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 15\left(-60\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-\left(-25\right)±\sqrt{625-60\left(-60\right)}}{2\times 15}$$

Whakareatia -60 ki te -60.

$$x=\frac{-\left(-25\right)±\sqrt{625+3600}}{2\times 15}$$

Tāpiri 625 ki te 3600.

$$x=\frac{-\left(-25\right)±\sqrt{4225}}{2\times 15}$$

Tuhia te pūtakerua o te 4225.

$$x=\frac{-\left(-25\right)±65}{2\times 15}$$

Ko te tauaro o -25 ko 25.

$$x=\frac{25±65}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{25±65}{30}$$

Nā, me whakaoti te whārite x=\frac{25±65}{30} ina he tāpiri te ±. Tāpiri 25 ki te 65.

$$x=\frac{90}{30}$$

Whakawehe 90 ki te 30.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{25±65}{30} ina he tango te ±. Tango 65 mai i 25.

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

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

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

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

$$15x^{2}-25x-60=15\left(x-3\right)\left(x-\left(-\frac{4}{3}\right)\right)$$

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

$$15x^{2}-25x-60=15\left(x-3\right)\left(x+\frac{4}{3}\right)$$

Tāpiri \frac{4}{3} 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.

$$15x^{2}-25x-60=15\left(x-3\right)\times \frac{3x+4}{3}$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te 15 me te 3.

$$15x^{2}-25x-60=5\left(x-3\right)\left(3x+4\right)$$

Aromātai

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