Whakaoti i te 3x^2-2x-1=0

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

Whakaoti mō x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$a+b=-2 ab=3\left(-1\right)=-3$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-3 b=1$$

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

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

Whakatauwehea atu 3x i te 3x^{2}-3x.

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

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

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

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 3x+1=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -2 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -2.

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

Whakareatia -4 ki te 3.

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

Whakareatia -12 ki te -1.

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

Tāpiri 4 ki te 12.

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

Tuhia te pūtakerua o te 16.

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

Ko te tauaro o -2 ko 2.

$$x=\frac{2±4}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{2±4}{6}$$

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

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

Whakawehe 6 ki te 6.

$$x=1$$

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

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

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

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

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

Me tāpiri 1 ki ngā taha e rua o te whārite.

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

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

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

Tango -1 mai i 0.

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

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-2x}{3}=\frac{1}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{2}{3}x=\frac{1}{3}$$

Whakawehea te -\frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{3}. Nā, tāpiria te pūrua o te -\frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{1}{3}+\frac{1}{9}$$

Tāpiri \frac{1}{3} ki te \frac{1}{9} 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.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{4}{9}$$

Tauwehea x^{2}-\frac{2}{3}x+\frac{1}{9}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{4}{9}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{4}{9}}$$

Whakarūnātia.

$$x-\frac{1}{3}=\frac{2}{3} x-\frac{1}{3}=-\frac{2}{3}$$

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

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