Whakaoti i te 3x^2-10x-8=0

$$3 { x }^{ 2 } -10x-8=0$$

Whakaoti mō x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=4$$

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-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=3\left(-8\right)=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -10.

$$a=-12 b=2$$

Tuhia anō te 3x^{2}-10x-8 hei \left(3x^{2}-12x\right)+\left(2x-8\right).

$$\left(3x^{2}-12x\right)+\left(2x-8\right)$$

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

$$3x\left(x-4\right)+2\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(3x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te 3x+2=0.

$$x=4 x=-\frac{2}{3}$$

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.

$$3x^{2}-10x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -10 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 3\left(-8\right)}}{2\times 3}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 3\left(-8\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-10\right)±\sqrt{100-12\left(-8\right)}}{2\times 3}$$

Whakareatia -12 ki te -8.

$$x=\frac{-\left(-10\right)±\sqrt{100+96}}{2\times 3}$$

Tāpiri 100 ki te 96.

$$x=\frac{-\left(-10\right)±\sqrt{196}}{2\times 3}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-\left(-10\right)±14}{2\times 3}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±14}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{10±14}{6}$$

Nā, me whakaoti te whārite x=\frac{10±14}{6} ina he tāpiri te ±. Tāpiri 10 ki te 14.

$$x=\frac{24}{6}$$

Whakawehe 24 ki te 6.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{10±14}{6} ina he tango te ±. Tango 14 mai i 10.

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

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

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

Kua oti te whārite te whakatau.

$$x=4 x=-\frac{2}{3}$$

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.

$$3x^{2}-10x-8=0$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$3x^{2}-10x-8-\left(-8\right)=-\left(-8\right)$$

Mā te tango i te -8 i a ia ake anō ka toe ko te 0.

$$3x^{2}-10x=-\left(-8\right)$$

Tango -8 mai i 0.

$$3x^{2}-10x=8$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-10x}{3}=\frac{8}{3}$$

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

$$x^{2}-\frac{10}{3}x=\frac{8}{3}$$

Whakawehea te -\frac{10}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{3}. Nā, tāpiria te pūrua o te -\frac{5}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{10}{3}x+\left(-\frac{5}{3}\right)^{2}=\frac{8}{3}+\left(-\frac{5}{3}\right)^{2}$$

Pūruatia -\frac{5}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=\frac{8}{3}+\frac{25}{9}$$

Tāpiri \frac{8}{3} ki te \frac{25}{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.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=\frac{49}{9}$$

Tauwehea te x^{2}-\frac{10}{3}x+\frac{25}{9}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{3}\right)^{2}=\frac{49}{9}$$

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

$$\sqrt{\left(x-\frac{5}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Whakarūnātia.

$$x-\frac{5}{3}=\frac{7}{3} x-\frac{5}{3}=-\frac{7}{3}$$

Me tāpiri \frac{5}{3} ki ngā taha e rua o te whārite.

$$x=4 x=-\frac{2}{3}$$