Whakaoti i te -14x^2+133x-63

$$- 14 x ^ { 2 } + 133 x - 63$$

Tauwehe

$$7\left(9-x\right)\left(2x-1\right)$$

Ngā Upane Otinga

Tauwehea te 7.

$$7\left(-2x^{2}+19x-9\right)$$

Whakaarohia te -2x^{2}+19x-9. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -2x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=19 ab=-2\left(-9\right)=18$$

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 18.

$$1,18 2,9 3,6$$

Tātaihia te tapeke mō ia takirua.

$$1+18=19 2+9=11 3+6=9$$

Ko te otinga te takirua ka hoatu i te tapeke 19.

$$a=18 b=1$$

Tuhia anō te -2x^{2}+19x-9 hei \left(-2x^{2}+18x\right)+\left(x-9\right).

$$\left(-2x^{2}+18x\right)+\left(x-9\right)$$

Tauwehea te 2x i te tuatahi me te -1 i te rōpū tuarua.

$$2x\left(-x+9\right)-\left(-x+9\right)$$

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

$$\left(-x+9\right)\left(2x-1\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$7\left(-x+9\right)\left(2x-1\right)$$

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.

$$-14x^{2}+133x-63=0$$

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.

$$x=\frac{-133±\sqrt{133^{2}-4\left(-14\right)\left(-63\right)}}{2\left(-14\right)}$$

Pūrua 133.

$$x=\frac{-133±\sqrt{17689-4\left(-14\right)\left(-63\right)}}{2\left(-14\right)}$$

Whakareatia -4 ki te -14.

$$x=\frac{-133±\sqrt{17689+56\left(-63\right)}}{2\left(-14\right)}$$

Whakareatia 56 ki te -63.

$$x=\frac{-133±\sqrt{17689-3528}}{2\left(-14\right)}$$

Tāpiri 17689 ki te -3528.

$$x=\frac{-133±\sqrt{14161}}{2\left(-14\right)}$$

Tuhia te pūtakerua o te 14161.

$$x=\frac{-133±119}{2\left(-14\right)}$$

Whakareatia 2 ki te -14.

$$x=\frac{-133±119}{-28}$$

Nā, me whakaoti te whārite x=\frac{-133±119}{-28} ina he tāpiri te ±. Tāpiri -133 ki te 119.

$$x=-\frac{14}{-28}$$

Whakahekea te hautanga \frac{-14}{-28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-133±119}{-28} ina he tango te ±. Tango 119 mai i -133.

$$x=-\frac{252}{-28}$$

Whakawehe -252 ki te -28.

$$x=9$$

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}{2} mō te x_{1} me te 9 mō te x_{2}.

$$-14x^{2}+133x-63=-14\left(x-\frac{1}{2}\right)\left(x-9\right)$$

Tango \frac{1}{2} 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.

$$-14x^{2}+133x-63=-14\times \frac{-2x+1}{-2}\left(x-9\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te -14 me te 2.

$$-14x^{2}+133x-63=7\left(-2x+1\right)\left(x-9\right)$$

Aromātai

$$-14x^{2}+133x-63$$