Whakaoti i te -117x^2+477x-36

$$-117 { x }^{ 2 } +477x-36$$

Tauwehe

$$9\left(4-x\right)\left(13x-1\right)$$

Ngā Upane Otinga

Tauwehea te 9.

$$9\left(-13x^{2}+53x-4\right)$$

Whakaarohia te -13x^{2}+53x-4. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -13x^{2}+ax+bx-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=53 ab=-13\left(-4\right)=52$$

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 52.

$$1,52 2,26 4,13$$

Tātaihia te tapeke mō ia takirua.

$$1+52=53 2+26=28 4+13=17$$

Ko te otinga te takirua ka hoatu i te tapeke 53.

$$a=52 b=1$$

Tuhia anō te -13x^{2}+53x-4 hei \left(-13x^{2}+52x\right)+\left(x-4\right).

$$\left(-13x^{2}+52x\right)+\left(x-4\right)$$

Tauwehea te 13x i te tuatahi me te -1 i te rōpū tuarua.

$$13x\left(-x+4\right)-\left(-x+4\right)$$

Whakatauwehea atu te kīanga pātahi -x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+4\right)\left(13x-1\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$9\left(-x+4\right)\left(13x-1\right)$$

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.

$$-117x^{2}+477x-36=0$$

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.

$$x=\frac{-477±\sqrt{477^{2}-4\left(-117\right)\left(-36\right)}}{2\left(-117\right)}$$

Pūrua 477.

$$x=\frac{-477±\sqrt{227529-4\left(-117\right)\left(-36\right)}}{2\left(-117\right)}$$

Whakareatia -4 ki te -117.

$$x=\frac{-477±\sqrt{227529+468\left(-36\right)}}{2\left(-117\right)}$$

Whakareatia 468 ki te -36.

$$x=\frac{-477±\sqrt{227529-16848}}{2\left(-117\right)}$$

Tāpiri 227529 ki te -16848.

$$x=\frac{-477±\sqrt{210681}}{2\left(-117\right)}$$

Tuhia te pūtakerua o te 210681.

$$x=\frac{-477±459}{2\left(-117\right)}$$

Whakareatia 2 ki te -117.

$$x=\frac{-477±459}{-234}$$

Nā, me whakaoti te whārite x=\frac{-477±459}{-234} ina he tāpiri te ±. Tāpiri -477 ki te 459.

$$x=-\frac{18}{-234}$$

Whakahekea te hautanga \frac{-18}{-234} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 18.

$$x=\frac{1}{13}$$

Nā, me whakaoti te whārite x=\frac{-477±459}{-234} ina he tango te ±. Tango 459 mai i -477.

$$x=-\frac{936}{-234}$$

Whakawehe -936 ki te -234.

$$x=4$$

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{1}{13} mō te x_{1} me te 4 mō te x_{2}.

$$-117x^{2}+477x-36=-117\left(x-\frac{1}{13}\right)\left(x-4\right)$$

Tango \frac{1}{13} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$-117x^{2}+477x-36=-117\times \frac{-13x+1}{-13}\left(x-4\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 13 i roto i te -117 me te 13.

$$-117x^{2}+477x-36=9\left(-13x+1\right)\left(x-4\right)$$

Aromātai

$$-117x^{2}+477x-36$$