Whakaoti i te 3x^2=4x+7

$$3 x ^ { 2 } = 4 x + 7$$

Whakaoti mō x

$$x=-1$$

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 4x mai i ngā taha e rua.

$$3x^{2}-4x=7$$

Tangohia te 7 mai i ngā taha e rua.

$$3x^{2}-4x-7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=3\left(-7\right)=-21$$

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 -21.

$$1,-21 3,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-21=-20 3-7=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-7 b=3$$

Tuhia anō te 3x^{2}-4x-7 hei \left(3x^{2}-7x\right)+\left(3x-7\right).

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

Whakatauwehea atu x i te 3x^{2}-7x.

$$x\left(3x-7\right)+3x-7$$

Whakatauwehea atu te kīanga pātahi 3x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-7\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-7=0 me te x+1=0.

$$x=\frac{7}{3} x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 4x mai i ngā taha e rua.

$$3x^{2}-4x=7$$

Tangohia te 7 mai i ngā taha e rua.

$$3x^{2}-4x-7=0$$

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

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

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-7\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\left(-7\right)}}{2\times 3}$$

Whakareatia -12 ki te -7.

$$x=\frac{-\left(-4\right)±\sqrt{16+84}}{2\times 3}$$

Tāpiri 16 ki te 84.

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

Tuhia te pūtakerua o te 100.

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

Ko te tauaro o -4 ko 4.

$$x=\frac{4±10}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{4±10}{6}$$

Nā, me whakaoti te whārite x=\frac{4±10}{6} ina he tāpiri te ±. Tāpiri 4 ki te 10.

$$x=\frac{14}{6}$$

Whakahekea te hautanga \frac{14}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{4±10}{6} ina he tango te ±. Tango 10 mai i 4.

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

Whakawehe -6 ki te 6.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=\frac{7}{3} x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 4x mai i ngā taha e rua.

$$3x^{2}-4x=7$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-4x}{3}=\frac{7}{3}$$

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

$$x^{2}-\frac{4}{3}x=\frac{7}{3}$$

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

$$x^{2}-\frac{4}{3}x+\left(-\frac{2}{3}\right)^{2}=\frac{7}{3}+\left(-\frac{2}{3}\right)^{2}$$

Pūruatia -\frac{2}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{7}{3}+\frac{4}{9}$$

Tāpiri \frac{7}{3} ki te \frac{4}{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.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{25}{9}$$

Tauwehea x^{2}-\frac{4}{3}x+\frac{4}{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}.

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

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

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

Whakarūnātia.

$$x-\frac{2}{3}=\frac{5}{3} x-\frac{2}{3}=-\frac{5}{3}$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$x=\frac{7}{3} x=-1$$