Whakaoti i te -116x^2+224x+16

$$- 116 x ^ { 2 } + 224 x + 16$$

Tauwehe

$$4\left(2-x\right)\left(29x+2\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(-29x^{2}+56x+4\right)$$

Whakaarohia te -29x^{2}+56x+4. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -29x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=56 ab=-29\times 4=-116$$

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 -116.

$$-1,116 -2,58 -4,29$$

Tātaihia te tapeke mō ia takirua.

$$-1+116=115 -2+58=56 -4+29=25$$

Ko te otinga te takirua ka hoatu i te tapeke 56.

$$a=58 b=-2$$

Tuhia anō te -29x^{2}+56x+4 hei \left(-29x^{2}+58x\right)+\left(-2x+4\right).

$$\left(-29x^{2}+58x\right)+\left(-2x+4\right)$$

Tauwehea te 29x i te tuatahi me te 2 i te rōpū tuarua.

$$29x\left(-x+2\right)+2\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(29x+2\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(-x+2\right)\left(29x+2\right)$$

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.

$$-116x^{2}+224x+16=0$$

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.

$$x=\frac{-224±\sqrt{224^{2}-4\left(-116\right)\times 16}}{2\left(-116\right)}$$

Pūrua 224.

$$x=\frac{-224±\sqrt{50176-4\left(-116\right)\times 16}}{2\left(-116\right)}$$

Whakareatia -4 ki te -116.

$$x=\frac{-224±\sqrt{50176+464\times 16}}{2\left(-116\right)}$$

Whakareatia 464 ki te 16.

$$x=\frac{-224±\sqrt{50176+7424}}{2\left(-116\right)}$$

Tāpiri 50176 ki te 7424.

$$x=\frac{-224±\sqrt{57600}}{2\left(-116\right)}$$

Tuhia te pūtakerua o te 57600.

$$x=\frac{-224±240}{2\left(-116\right)}$$

Whakareatia 2 ki te -116.

$$x=\frac{-224±240}{-232}$$

Nā, me whakaoti te whārite x=\frac{-224±240}{-232} ina he tāpiri te ±. Tāpiri -224 ki te 240.

$$x=\frac{16}{-232}$$

Whakahekea te hautanga \frac{16}{-232} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=-\frac{2}{29}$$

Nā, me whakaoti te whārite x=\frac{-224±240}{-232} ina he tango te ±. Tango 240 mai i -224.

$$x=-\frac{464}{-232}$$

Whakawehe -464 ki te -232.

$$x=2$$

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 -\frac{2}{29} mō te x_{1} me te 2 mō te x_{2}.

$$-116x^{2}+224x+16=-116\left(x-\left(-\frac{2}{29}\right)\right)\left(x-2\right)$$

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

$$-116x^{2}+224x+16=-116\left(x+\frac{2}{29}\right)\left(x-2\right)$$

Tāpiri \frac{2}{29} ki te x mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-116x^{2}+224x+16=-116\times \frac{-29x-2}{-29}\left(x-2\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 29 i roto i te -116 me te 29.

$$-116x^{2}+224x+16=4\left(-29x-2\right)\left(x-2\right)$$

Aromātai

$$16+224x-116x^{2}$$