Whakaoti i te M(x)=-25x^2+100x+8000

$$M ( x ) = - 25 x ^ { 2 } + 100 x + 8000$$

Tauwehe

$$-25\left(x-20\right)\left(x+16\right)$$

Ngā Upane Otinga

Tauwehea te 25.

$$25\left(-x^{2}+4x+320\right)$$

Whakaarohia te -x^{2}+4x+320. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+320. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=-320=-320$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -320.

$$-1,320 -2,160 -4,80 -5,64 -8,40 -10,32 -16,20$$

Tātaihia te tapeke mō ia takirua.

$$-1+320=319 -2+160=158 -4+80=76 -5+64=59 -8+40=32 -10+32=22 -16+20=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=20 b=-16$$

Tuhia anō te -x^{2}+4x+320 hei \left(-x^{2}+20x\right)+\left(-16x+320\right).

$$\left(-x^{2}+20x\right)+\left(-16x+320\right)$$

Tauwehea te -x i te tuatahi me te -16 i te rōpū tuarua.

$$-x\left(x-20\right)-16\left(x-20\right)$$

Whakatauwehea atu te kīanga pātahi x-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-20\right)\left(-x-16\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$25\left(x-20\right)\left(-x-16\right)$$

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.

$$-25x^{2}+100x+8000=0$$

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.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-25\right)\times 8000}}{2\left(-25\right)}$$

Pūrua 100.

$$x=\frac{-100±\sqrt{10000-4\left(-25\right)\times 8000}}{2\left(-25\right)}$$

Whakareatia -4 ki te -25.

$$x=\frac{-100±\sqrt{10000+100\times 8000}}{2\left(-25\right)}$$

Whakareatia 100 ki te 8000.

$$x=\frac{-100±\sqrt{10000+800000}}{2\left(-25\right)}$$

Tāpiri 10000 ki te 800000.

$$x=\frac{-100±\sqrt{810000}}{2\left(-25\right)}$$

Tuhia te pūtakerua o te 810000.

$$x=\frac{-100±900}{2\left(-25\right)}$$

Whakareatia 2 ki te -25.

$$x=\frac{-100±900}{-50}$$

Nā, me whakaoti te whārite x=\frac{-100±900}{-50} ina he tāpiri te ±. Tāpiri -100 ki te 900.

$$x=\frac{800}{-50}$$

Whakawehe 800 ki te -50.

$$x=-16$$

Nā, me whakaoti te whārite x=\frac{-100±900}{-50} ina he tango te ±. Tango 900 mai i -100.

$$x=-\frac{1000}{-50}$$

Whakawehe -1000 ki te -50.

$$x=20$$

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 -16 mō te x_{1} me te 20 mō te x_{2}.

$$-25x^{2}+100x+8000=-25\left(x-\left(-16\right)\right)\left(x-20\right)$$

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

$$-25x^{2}+100x+8000=-25\left(x+16\right)\left(x-20\right)$$

Aromātai

$$-25\left(x-20\right)\left(x+16\right)$$