Whakaoti i te x^2+20x+99

$$x ^ { 2 } + 20 x + 99$$

Tauwehe

$$\left(x+9\right)\left(x+11\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+99. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=20 ab=1\times 99=99$$

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 99.

$$1,99 3,33 9,11$$

Tātaihia te tapeke mō ia takirua.

$$1+99=100 3+33=36 9+11=20$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=9 b=11$$

Tuhia anō te x^{2}+20x+99 hei \left(x^{2}+9x\right)+\left(11x+99\right).

$$\left(x^{2}+9x\right)+\left(11x+99\right)$$

Tauwehea te x i te tuatahi me te 11 i te rōpū tuarua.

$$x\left(x+9\right)+11\left(x+9\right)$$

Whakatauwehea atu te kīanga pātahi x+9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+9\right)\left(x+11\right)$$

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.

$$x^{2}+20x+99=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\times 99}}{2}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\times 99}}{2}$$

Whakareatia -4 ki te 99.

$$x=\frac{-20±\sqrt{400-396}}{2}$$

Tāpiri 400 ki te -396.

$$x=\frac{-20±\sqrt{4}}{2}$$

Tuhia te pūtakerua o te 4.

$$x=\frac{-20±2}{2}$$

Nā, me whakaoti te whārite x=\frac{-20±2}{2} ina he tāpiri te ±. Tāpiri -20 ki te 2.

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

Whakawehe -18 ki te 2.

$$x=-9$$

Nā, me whakaoti te whārite x=\frac{-20±2}{2} ina he tango te ±. Tango 2 mai i -20.

$$x=-\frac{22}{2}$$

Whakawehe -22 ki te 2.

$$x=-11$$

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 -11 mō te x_{2}.

$$x^{2}+20x+99=\left(x-\left(-9\right)\right)\left(x-\left(-11\right)\right)$$

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

$$x^{2}+20x+99=\left(x+9\right)\left(x+11\right)$$

Aromātai

$$\left(x+9\right)\left(x+11\right)$$