Whakaoti i te 15x^2-26x-57

$$15 { x }^{ 2 } -26x-57$$

Tauwehe

$$\left(x-3\right)\left(15x+19\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 15x^{2}+ax+bx-57. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-26 ab=15\left(-57\right)=-855$$

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 -855.

$$1,-855 3,-285 5,-171 9,-95 15,-57 19,-45$$

Tātaihia te tapeke mō ia takirua.

$$1-855=-854 3-285=-282 5-171=-166 9-95=-86 15-57=-42 19-45=-26$$

Ko te otinga te takirua ka hoatu i te tapeke -26.

$$a=-45 b=19$$

Tuhia anō te 15x^{2}-26x-57 hei \left(15x^{2}-45x\right)+\left(19x-57\right).

$$\left(15x^{2}-45x\right)+\left(19x-57\right)$$

Tauwehea te 15x i te tuatahi me te 19 i te rōpū tuarua.

$$15x\left(x-3\right)+19\left(x-3\right)$$

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

$$\left(x-3\right)\left(15x+19\right)$$

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.

$$15x^{2}-26x-57=0$$

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.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 15\left(-57\right)}}{2\times 15}$$

Pūrua -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 15\left(-57\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-\left(-26\right)±\sqrt{676-60\left(-57\right)}}{2\times 15}$$

Whakareatia -60 ki te -57.

$$x=\frac{-\left(-26\right)±\sqrt{676+3420}}{2\times 15}$$

Tāpiri 676 ki te 3420.

$$x=\frac{-\left(-26\right)±\sqrt{4096}}{2\times 15}$$

Tuhia te pūtakerua o te 4096.

$$x=\frac{-\left(-26\right)±64}{2\times 15}$$

Ko te tauaro o -26 ko 26.

$$x=\frac{26±64}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{26±64}{30}$$

Nā, me whakaoti te whārite x=\frac{26±64}{30} ina he tāpiri te ±. Tāpiri 26 ki te 64.

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

Whakawehe 90 ki te 30.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{26±64}{30} ina he tango te ±. Tango 64 mai i 26.

$$x=-\frac{38}{30}$$

Whakahekea te hautanga \frac{-38}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{19}{15}$$

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{19}{15} mō te x_{2}.

$$15x^{2}-26x-57=15\left(x-3\right)\left(x-\left(-\frac{19}{15}\right)\right)$$

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

$$15x^{2}-26x-57=15\left(x-3\right)\left(x+\frac{19}{15}\right)$$

Tāpiri \frac{19}{15} 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.

$$15x^{2}-26x-57=15\left(x-3\right)\times \frac{15x+19}{15}$$

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

$$15x^{2}-26x-57=\left(x-3\right)\left(15x+19\right)$$

Aromātai

$$\left(x-3\right)\left(15x+19\right)$$