Whakaoti i te -3x^2-4x-1

$$-3 { x }^{ 2 } -4x-1$$

Tauwehe

$$-\left(x+1\right)\left(3x+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -3x^{2}+ax+bx-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=-3\left(-1\right)=3$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-1 b=-3$$

Tuhia anō te -3x^{2}-4x-1 hei \left(-3x^{2}-x\right)+\left(-3x-1\right).

$$\left(-3x^{2}-x\right)+\left(-3x-1\right)$$

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

$$-x\left(3x+1\right)-\left(3x+1\right)$$

Whakatauwehea atu te kīanga pātahi 3x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+1\right)\left(-x-1\right)$$

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.

$$-3x^{2}-4x-1=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-4\right)±\sqrt{16+12\left(-1\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16-12}}{2\left(-3\right)}$$

Tāpiri 16 ki te -12.

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

Tuhia te pūtakerua o te 4.

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

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{4±2}{-6}$$

Nā, me whakaoti te whārite x=\frac{4±2}{-6} ina he tāpiri te ±. Tāpiri 4 ki te 2.

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

Whakawehe 6 ki te -6.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{4±2}{-6} ina he tango te ±. Tango 2 mai i 4.

$$x=\frac{2}{-6}$$

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

$$x=-\frac{1}{3}$$

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 -1 mō te x_{1} me te -\frac{1}{3} mō te x_{2}.

$$-3x^{2}-4x-1=-3\left(x-\left(-1\right)\right)\left(x-\left(-\frac{1}{3}\right)\right)$$

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

$$-3x^{2}-4x-1=-3\left(x+1\right)\left(x+\frac{1}{3}\right)$$

Tāpiri \frac{1}{3} 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.

$$-3x^{2}-4x-1=-3\left(x+1\right)\times \frac{-3x-1}{-3}$$

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

$$-3x^{2}-4x-1=\left(x+1\right)\left(-3x-1\right)$$

Aromātai

$$-\left(x+1\right)\left(3x+1\right)$$