Whakaoti i te -3x^2+38x-24

$$-3 { x }^{ 2 } +38x-24$$

Tauwehe

$$\left(12-x\right)\left(3x-2\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -3x^{2}+ax+bx-24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=38 ab=-3\left(-24\right)=72$$

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 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Tātaihia te tapeke mō ia takirua.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

Ko te otinga te takirua ka hoatu i te tapeke 38.

$$a=36 b=2$$

Tuhia anō te -3x^{2}+38x-24 hei \left(-3x^{2}+36x\right)+\left(2x-24\right).

$$\left(-3x^{2}+36x\right)+\left(2x-24\right)$$

Tauwehea te 3x i te tuatahi me te -2 i te rōpū tuarua.

$$3x\left(-x+12\right)-2\left(-x+12\right)$$

Whakatauwehea atu te kīanga pātahi -x+12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+12\right)\left(3x-2\right)$$

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.

$$-3x^{2}+38x-24=0$$

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.

$$x=\frac{-38±\sqrt{38^{2}-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Pūrua 38.

$$x=\frac{-38±\sqrt{1444-4\left(-3\right)\left(-24\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-38±\sqrt{1444+12\left(-24\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -24.

$$x=\frac{-38±\sqrt{1444-288}}{2\left(-3\right)}$$

Tāpiri 1444 ki te -288.

$$x=\frac{-38±\sqrt{1156}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 1156.

$$x=\frac{-38±34}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-38±34}{-6}$$

Nā, me whakaoti te whārite x=\frac{-38±34}{-6} ina he tāpiri te ±. Tāpiri -38 ki te 34.

$$x=-\frac{4}{-6}$$

Whakahekea te hautanga \frac{-4}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{2}{3}$$

Nā, me whakaoti te whārite x=\frac{-38±34}{-6} ina he tango te ±. Tango 34 mai i -38.

$$x=-\frac{72}{-6}$$

Whakawehe -72 ki te -6.

$$x=12$$

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}{3} mō te x_{1} me te 12 mō te x_{2}.

$$-3x^{2}+38x-24=-3\left(x-\frac{2}{3}\right)\left(x-12\right)$$

Tango \frac{2}{3} 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.

$$-3x^{2}+38x-24=-3\times \frac{-3x+2}{-3}\left(x-12\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te -3 me te 3.

$$-3x^{2}+38x-24=\left(-3x+2\right)\left(x-12\right)$$

Aromātai

$$\left(12-x\right)\left(3x-2\right)$$