Whakaoti i te -5u^2+45u-90

$$- 5 u ^ { 2 } + 45 u - 90$$

Tauwehe

$$5\left(3-u\right)\left(u-6\right)$$

Ngā Upane Otinga

Tauwehea te 5.

$$5\left(-u^{2}+9u-18\right)$$

Whakaarohia te -u^{2}+9u-18. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -u^{2}+au+bu-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=-\left(-18\right)=18$$

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 18.

$$1,18 2,9 3,6$$

Tātaihia te tapeke mō ia takirua.

$$1+18=19 2+9=11 3+6=9$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=6 b=3$$

Tuhia anō te -u^{2}+9u-18 hei \left(-u^{2}+6u\right)+\left(3u-18\right).

$$\left(-u^{2}+6u\right)+\left(3u-18\right)$$

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

$$-u\left(u-6\right)+3\left(u-6\right)$$

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

$$\left(u-6\right)\left(-u+3\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$5\left(u-6\right)\left(-u+3\right)$$

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.

$$-5u^{2}+45u-90=0$$

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.

$$u=\frac{-45±\sqrt{45^{2}-4\left(-5\right)\left(-90\right)}}{2\left(-5\right)}$$

Pūrua 45.

$$u=\frac{-45±\sqrt{2025-4\left(-5\right)\left(-90\right)}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$u=\frac{-45±\sqrt{2025+20\left(-90\right)}}{2\left(-5\right)}$$

Whakareatia 20 ki te -90.

$$u=\frac{-45±\sqrt{2025-1800}}{2\left(-5\right)}$$

Tāpiri 2025 ki te -1800.

$$u=\frac{-45±\sqrt{225}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te 225.

$$u=\frac{-45±15}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$u=\frac{-45±15}{-10}$$

Nā, me whakaoti te whārite u=\frac{-45±15}{-10} ina he tāpiri te ±. Tāpiri -45 ki te 15.

$$u=-\frac{30}{-10}$$

Whakawehe -30 ki te -10.

$$u=3$$

Nā, me whakaoti te whārite u=\frac{-45±15}{-10} ina he tango te ±. Tango 15 mai i -45.

$$u=-\frac{60}{-10}$$

Whakawehe -60 ki te -10.

$$u=6$$

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 3 mō te x_{1} me te 6 mō te x_{2}.

$$-5u^{2}+45u-90=-5\left(u-3\right)\left(u-6\right)$$

Aromātai

$$5\left(3-u\right)\left(u-6\right)$$