Whakaoti i te -20a-100-a^2=

$$- 20 a - 100 - a ^ { 2 } =$$

Tauwehe

$$-\left(a+10\right)^{2}$$

Ngā Upane Otinga

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-a^{2}-20a-100$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -a^{2}+pa+qa-100. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-20 pq=-\left(-100\right)=100$$

I te mea kua tōrunga te pq, he ōrite te tohu o p me q. I te mea kua tōraro te p+q, he tōraro hoki a p me q. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 100.

$$-1,-100 -2,-50 -4,-25 -5,-20 -10,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-100=-101 -2-50=-52 -4-25=-29 -5-20=-25 -10-10=-20$$

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

$$p=-10 q=-10$$

Tuhia anō te -a^{2}-20a-100 hei \left(-a^{2}-10a\right)+\left(-10a-100\right).

$$\left(-a^{2}-10a\right)+\left(-10a-100\right)$$

Tauwehea te -a i te tuatahi me te -10 i te rōpū tuarua.

$$-a\left(a+10\right)-10\left(a+10\right)$$

Whakatauwehea atu te kīanga pātahi a+10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(a+10\right)\left(-a-10\right)$$

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.

$$-a^{2}-20a-100=0$$

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.

$$a=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-1\right)\left(-100\right)}}{2\left(-1\right)}$$

Pūrua -20.

$$a=\frac{-\left(-20\right)±\sqrt{400-4\left(-1\right)\left(-100\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$a=\frac{-\left(-20\right)±\sqrt{400+4\left(-100\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -100.

$$a=\frac{-\left(-20\right)±\sqrt{400-400}}{2\left(-1\right)}$$

Tāpiri 400 ki te -400.

$$a=\frac{-\left(-20\right)±\sqrt{0}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 0.

$$a=\frac{-\left(-20\right)±0}{2\left(-1\right)}$$

Ko te tauaro o -20 ko 20.

$$a=\frac{20±0}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$a=\frac{20±0}{-2}$$

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 -10 mō te x_{1} me te -10 mō te x_{2}.

$$-a^{2}-20a-100=-\left(a-\left(-10\right)\right)\left(a-\left(-10\right)\right)$$

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

$$-a^{2}-20a-100=-\left(a+10\right)\left(a+10\right)$$

Aromātai

$$-\left(a+10\right)^{2}$$