Whakaoti i te -x^2-3x+54

$$- x ^ { 2 } - 3 x + 54$$

Tauwehe

$$\left(6-x\right)\left(x+9\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+54. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=-54=-54$$

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 -54.

$$1,-54 2,-27 3,-18 6,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-54=-53 2-27=-25 3-18=-15 6-9=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=6 b=-9$$

Tuhia anō te -x^{2}-3x+54 hei \left(-x^{2}+6x\right)+\left(-9x+54\right).

$$\left(-x^{2}+6x\right)+\left(-9x+54\right)$$

Tauwehea te x i te tuatahi me te 9 i te rōpū tuarua.

$$x\left(-x+6\right)+9\left(-x+6\right)$$

Whakatauwehea atu te kīanga pātahi -x+6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+6\right)\left(x+9\right)$$

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.

$$-x^{2}-3x+54=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 54}}{2\left(-1\right)}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 54}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 54}}{2\left(-1\right)}$$

Whakareatia 4 ki te 54.

$$x=\frac{-\left(-3\right)±\sqrt{9+216}}{2\left(-1\right)}$$

Tāpiri 9 ki te 216.

$$x=\frac{-\left(-3\right)±\sqrt{225}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 225.

$$x=\frac{-\left(-3\right)±15}{2\left(-1\right)}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±15}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{3±15}{-2}$$

Nā, me whakaoti te whārite x=\frac{3±15}{-2} ina he tāpiri te ±. Tāpiri 3 ki te 15.

$$x=\frac{18}{-2}$$

Whakawehe 18 ki te -2.

$$x=-9$$

Nā, me whakaoti te whārite x=\frac{3±15}{-2} ina he tango te ±. Tango 15 mai i 3.

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

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 -9 mō te x_{1} me te 6 mō te x_{2}.

$$-x^{2}-3x+54=-\left(x-\left(-9\right)\right)\left(x-6\right)$$

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

$$-x^{2}-3x+54=-\left(x+9\right)\left(x-6\right)$$

Aromātai

$$\left(6-x\right)\left(x+9\right)$$