Whakaoti i te 3x^2-8x-3=0

$$3 { x }^{ 2 } -8x-3=0$$

Whakaoti mō x

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

$$x=3$$

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-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=3\left(-3\right)=-9$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -9.

$$1,-9 3,-3$$

Tātaihia te tapeke mō ia takirua.

$$1-9=-8 3-3=0$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-9 b=1$$

Tuhia anō te 3x^{2}-8x-3 hei \left(3x^{2}-9x\right)+\left(x-3\right).

$$\left(3x^{2}-9x\right)+\left(x-3\right)$$

Whakatauwehea atu 3x i te 3x^{2}-9x.

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

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

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

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

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

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.

$$3x^{2}-8x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -8 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3\left(-3\right)}}{2\times 3}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 3\left(-3\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-8\right)±\sqrt{64-12\left(-3\right)}}{2\times 3}$$

Whakareatia -12 ki te -3.

$$x=\frac{-\left(-8\right)±\sqrt{64+36}}{2\times 3}$$

Tāpiri 64 ki te 36.

$$x=\frac{-\left(-8\right)±\sqrt{100}}{2\times 3}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-8\right)±10}{2\times 3}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±10}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{8±10}{6}$$

Nā, me whakaoti te whārite x=\frac{8±10}{6} ina he tāpiri te ±. Tāpiri 8 ki te 10.

$$x=\frac{18}{6}$$

Whakawehe 18 ki te 6.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{8±10}{6} ina he tango te ±. Tango 10 mai i 8.

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

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.

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

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.

$$3x^{2}-8x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$3x^{2}-8x-3-\left(-3\right)=-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$3x^{2}-8x=-\left(-3\right)$$

Tango -3 mai i 0.

$$3x^{2}-8x=3$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-8x}{3}=\frac{3}{3}$$

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

$$x^{2}-\frac{8}{3}x=\frac{3}{3}$$

Whakawehe 3 ki te 3.

$$x^{2}-\frac{8}{3}x=1$$

Whakawehea te -\frac{8}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4}{3}. Nā, tāpiria te pūrua o te -\frac{4}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=1+\left(-\frac{4}{3}\right)^{2}$$

Pūruatia -\frac{4}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=1+\frac{16}{9}$$

Tāpiri 1 ki te \frac{16}{9}.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{25}{9}$$

Tauwehea x^{2}-\frac{8}{3}x+\frac{16}{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}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{25}{9}$$

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

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{25}{9}}$$

Whakarūnātia.

$$x-\frac{4}{3}=\frac{5}{3} x-\frac{4}{3}=-\frac{5}{3}$$

Me tāpiri \frac{4}{3} ki ngā taha e rua o te whārite.

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