Whakaoti i te x^2-12x-45=

$$x ^ { 2 } - 12 x - 45 =$$

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-12 ab=1\left(-45\right)=-45$$

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 -45.

$$1,-45 3,-15 5,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-45=-44 3-15=-12 5-9=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-15 b=3$$

Tuhia anō te x^{2}-12x-45 hei \left(x^{2}-15x\right)+\left(3x-45\right).

$$\left(x^{2}-15x\right)+\left(3x-45\right)$$

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

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

Whakatauwehea atu te kīanga pātahi x-15 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

$$x^{2}-12x-45=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-45\right)}}{2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-45\right)}}{2}$$

Whakareatia -4 ki te -45.

$$x=\frac{-\left(-12\right)±\sqrt{144+180}}{2}$$

Tāpiri 144 ki te 180.

$$x=\frac{-\left(-12\right)±\sqrt{324}}{2}$$

Tuhia te pūtakerua o te 324.

$$x=\frac{-\left(-12\right)±18}{2}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±18}{2}$$

Nā, me whakaoti te whārite x=\frac{12±18}{2} ina he tāpiri te ±. Tāpiri 12 ki te 18.

$$x=\frac{30}{2}$$

Whakawehe 30 ki te 2.

$$x=15$$

Nā, me whakaoti te whārite x=\frac{12±18}{2} ina he tango te ±. Tango 18 mai i 12.

$$x=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$x=-3$$

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 15 mō te x_{1} me te -3 mō te x_{2}.

$$x^{2}-12x-45=\left(x-15\right)\left(x-\left(-3\right)\right)$$

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

$$x^{2}-12x-45=\left(x-15\right)\left(x+3\right)$$

Aromātai

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