Whakaoti i te 3u^2+51u+90

$$3 u ^ { 2 } + 51 u + 90$$

Tauwehe

$$3\left(u+2\right)\left(u+15\right)$$

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(u^{2}+17u+30\right)$$

Whakaarohia te u^{2}+17u+30. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei u^{2}+au+bu+30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=1\times 30=30$$

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 30.

$$1,30 2,15 3,10 5,6$$

Tātaihia te tapeke mō ia takirua.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

Ko te otinga te takirua ka hoatu i te tapeke 17.

$$a=2 b=15$$

Tuhia anō te u^{2}+17u+30 hei \left(u^{2}+2u\right)+\left(15u+30\right).

$$\left(u^{2}+2u\right)+\left(15u+30\right)$$

Tauwehea te u i te tuatahi me te 15 i te rōpū tuarua.

$$u\left(u+2\right)+15\left(u+2\right)$$

Whakatauwehea atu te kīanga pātahi u+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(u+2\right)\left(u+15\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$3\left(u+2\right)\left(u+15\right)$$

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.

$$3u^{2}+51u+90=0$$

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.

$$u=\frac{-51±\sqrt{51^{2}-4\times 3\times 90}}{2\times 3}$$

Pūrua 51.

$$u=\frac{-51±\sqrt{2601-4\times 3\times 90}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$u=\frac{-51±\sqrt{2601-12\times 90}}{2\times 3}$$

Whakareatia -12 ki te 90.

$$u=\frac{-51±\sqrt{2601-1080}}{2\times 3}$$

Tāpiri 2601 ki te -1080.

$$u=\frac{-51±\sqrt{1521}}{2\times 3}$$

Tuhia te pūtakerua o te 1521.

$$u=\frac{-51±39}{2\times 3}$$

Whakareatia 2 ki te 3.

$$u=\frac{-51±39}{6}$$

Nā, me whakaoti te whārite u=\frac{-51±39}{6} ina he tāpiri te ±. Tāpiri -51 ki te 39.

$$u=-\frac{12}{6}$$

Whakawehe -12 ki te 6.

$$u=-2$$

Nā, me whakaoti te whārite u=\frac{-51±39}{6} ina he tango te ±. Tango 39 mai i -51.

$$u=-\frac{90}{6}$$

Whakawehe -90 ki te 6.

$$u=-15$$

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 -2 mō te x_{1} me te -15 mō te x_{2}.

$$3u^{2}+51u+90=3\left(u-\left(-2\right)\right)\left(u-\left(-15\right)\right)$$

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

$$3u^{2}+51u+90=3\left(u+2\right)\left(u+15\right)$$

Aromātai

$$3\left(u+2\right)\left(u+15\right)$$