Whakaoti i te -2x^2+12x-18

$$-2 { x }^{ 2 } +12x-18$$

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 2.

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

Whakaarohia te -x^{2}+6x-9. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=-\left(-9\right)=9$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

$$1,9 3,3$$

Tātaihia te tapeke mō ia takirua.

$$1+9=10 3+3=6$$

Ko te otinga te takirua ka hoatu i te tapeke 6.

$$a=3 b=3$$

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

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

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

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

$$2\left(x-3\right)\left(-x+3\right)$$

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.

$$-2x^{2}+12x-18=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-2\right)\left(-18\right)}}{2\left(-2\right)}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\left(-2\right)\left(-18\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-12±\sqrt{144+8\left(-18\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -18.

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

Tāpiri 144 ki te -144.

$$x=\frac{-12±\sqrt{0}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-12±0}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-12±0}{-4}$$

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 3 mō te x_{2}.

$$-2x^{2}+12x-18=-2\left(x-3\right)\left(x-3\right)$$

Aromātai

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