Whakaoti i te 3x^2-7x=6

$$3 x ^ { 2 } - 7 x = 6$$

Whakaoti mō x

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

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 6 mai i ngā taha e rua.

$$3x^{2}-7x-6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=3\left(-6\right)=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-18=-17 2-9=-7 3-6=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-9 b=2$$

Tuhia anō te 3x^{2}-7x-6 hei \left(3x^{2}-9x\right)+\left(2x-6\right).

$$\left(3x^{2}-9x\right)+\left(2x-6\right)$$

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

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

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

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

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

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

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.

$$3x^{2}-7x=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x^{2}-7x-6=6-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$3x^{2}-7x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -7 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\left(-6\right)}}{2\times 3}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3\left(-6\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-12\left(-6\right)}}{2\times 3}$$

Whakareatia -12 ki te -6.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2\times 3}$$

Tāpiri 49 ki te 72.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2\times 3}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-\left(-7\right)±11}{2\times 3}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±11}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{7±11}{6}$$

Nā, me whakaoti te whārite x=\frac{7±11}{6} ina he tāpiri te ±. Tāpiri 7 ki te 11.

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

Whakawehe 18 ki te 6.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{7±11}{6} ina he tango te ±. Tango 11 mai i 7.

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

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.

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

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.

$$3x^{2}-7x=6$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-7x}{3}=\frac{6}{3}$$

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

$$x^{2}-\frac{7}{3}x=\frac{6}{3}$$

Whakawehe 6 ki te 3.

$$x^{2}-\frac{7}{3}x=2$$

Whakawehea te -\frac{7}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{6}. Nā, tāpiria te pūrua o te -\frac{7}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=2+\left(-\frac{7}{6}\right)^{2}$$

Pūruatia -\frac{7}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=2+\frac{49}{36}$$

Tāpiri 2 ki te \frac{49}{36}.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{121}{36}$$

Tauwehea x^{2}-\frac{7}{3}x+\frac{49}{36}. 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}.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{121}{36}$$

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

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Whakarūnātia.

$$x-\frac{7}{6}=\frac{11}{6} x-\frac{7}{6}=-\frac{11}{6}$$

Me tāpiri \frac{7}{6} ki ngā taha e rua o te whārite.

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