Whakaoti i te x^2-2x-48

$${ x }^{ 2 } -2x-48$$

Tauwehe

$$\left(x-8\right)\left(x+6\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-48. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-48\right)=-48$$

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 -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-8 b=6$$

Tuhia anō te x^{2}-2x-48 hei \left(x^{2}-8x\right)+\left(6x-48\right).

$$\left(x^{2}-8x\right)+\left(6x-48\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-8\right)+6\left(x-8\right)$$

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

$$\left(x-8\right)\left(x+6\right)$$

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.

$$x^{2}-2x-48=0$$

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.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-48\right)}}{2}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-48\right)}}{2}$$

Whakareatia -4 ki te -48.

$$x=\frac{-\left(-2\right)±\sqrt{4+192}}{2}$$

Tāpiri 4 ki te 192.

$$x=\frac{-\left(-2\right)±\sqrt{196}}{2}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-\left(-2\right)±14}{2}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±14}{2}$$

Nā, me whakaoti te whārite x=\frac{2±14}{2} ina he tāpiri te ±. Tāpiri 2 ki te 14.

$$x=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{2±14}{2} ina he tango te ±. Tango 14 mai i 2.

$$x=-\frac{12}{2}$$

Whakawehe -12 ki te 2.

$$x=-6$$

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 8 mō te x_{1} me te -6 mō te x_{2}.

$$x^{2}-2x-48=\left(x-8\right)\left(x-\left(-6\right)\right)$$

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

$$x^{2}-2x-48=\left(x-8\right)\left(x+6\right)$$

Aromātai

$$\left(x-8\right)\left(x+6\right)$$