Whakaoti i te 3x^2+16x-12=0

$$3 x ^ { 2 } + 16 x - 12 = 0$$

Whakaoti mō x

$$x=-6$$

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

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-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=16 ab=3\left(-12\right)=-36$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -36.

$$-1,36 -2,18 -3,12 -4,9 -6,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0$$

Ko te otinga te takirua ka hoatu i te tapeke 16.

$$a=-2 b=18$$

Tuhia anō te 3x^{2}+16x-12 hei \left(3x^{2}-2x\right)+\left(18x-12\right).

$$\left(3x^{2}-2x\right)+\left(18x-12\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(3x-2\right)+6\left(3x-2\right)$$

Whakatauwehea atu te kīanga pātahi 3x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-2\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-2=0 me te x+6=0.

$$x=\frac{2}{3} x=-6$$

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.

$$3x^{2}+16x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 16 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\times 3\left(-12\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-16±\sqrt{256-12\left(-12\right)}}{2\times 3}$$

Whakareatia -12 ki te -12.

$$x=\frac{-16±\sqrt{256+144}}{2\times 3}$$

Tāpiri 256 ki te 144.

$$x=\frac{-16±\sqrt{400}}{2\times 3}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-16±20}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-16±20}{6}$$

Nā, me whakaoti te whārite x=\frac{-16±20}{6} ina he tāpiri te ±. Tāpiri -16 ki te 20.

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

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{-16±20}{6} ina he tango te ±. Tango 20 mai i -16.

$$x=-\frac{36}{6}$$

Whakawehe -36 ki te 6.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=\frac{2}{3} x=-6$$

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.

$$3x^{2}+16x-12=0$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$3x^{2}+16x-12-\left(-12\right)=-\left(-12\right)$$

Mā te tango i te -12 i a ia ake anō ka toe ko te 0.

$$3x^{2}+16x=-\left(-12\right)$$

Tango -12 mai i 0.

$$3x^{2}+16x=12$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+16x}{3}=\frac{12}{3}$$

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

$$x^{2}+\frac{16}{3}x=\frac{12}{3}$$

Whakawehe 12 ki te 3.

$$x^{2}+\frac{16}{3}x=4$$

Whakawehea te \frac{16}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{8}{3}. Nā, tāpiria te pūrua o te \frac{8}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=4+\left(\frac{8}{3}\right)^{2}$$

Pūruatia \frac{8}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=4+\frac{64}{9}$$

Tāpiri 4 ki te \frac{64}{9}.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{100}{9}$$

Tauwehea x^{2}+\frac{16}{3}x+\frac{64}{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}.

$$\left(x+\frac{8}{3}\right)^{2}=\frac{100}{9}$$

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

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{\frac{100}{9}}$$

Whakarūnātia.

$$x+\frac{8}{3}=\frac{10}{3} x+\frac{8}{3}=-\frac{10}{3}$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{2}{3} x=-6$$