Whakaoti i te 3x^2-4x=39

$$3 x ^ { 2 } - 4 x = 39$$

Whakaoti mō x

$$x=-3$$

$$x = \frac{13}{3} = 4\frac{1}{3} \approx 4.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 39 mai i ngā taha e rua.

$$3x^{2}-4x-39=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-39. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=3\left(-39\right)=-117$$

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 -117.

$$1,-117 3,-39 9,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-117=-116 3-39=-36 9-13=-4$$

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

$$a=-13 b=9$$

Tuhia anō te 3x^{2}-4x-39 hei \left(3x^{2}-13x\right)+\left(9x-39\right).

$$\left(3x^{2}-13x\right)+\left(9x-39\right)$$

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

$$x\left(3x-13\right)+3\left(3x-13\right)$$

Whakatauwehea atu te kīanga pātahi 3x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-13\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-13=0 me te x+3=0.

$$x=\frac{13}{3} x=-3$$

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.

$$3x^{2}-4x=39$$

Me tango 39 mai i ngā taha e rua o te whārite.

$$3x^{2}-4x-39=39-39$$

Mā te tango i te 39 i a ia ake anō ka toe ko te 0.

$$3x^{2}-4x-39=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -4 mō b, me -39 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-39\right)}}{2\times 3}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-39\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\left(-39\right)}}{2\times 3}$$

Whakareatia -12 ki te -39.

$$x=\frac{-\left(-4\right)±\sqrt{16+468}}{2\times 3}$$

Tāpiri 16 ki te 468.

$$x=\frac{-\left(-4\right)±\sqrt{484}}{2\times 3}$$

Tuhia te pūtakerua o te 484.

$$x=\frac{-\left(-4\right)±22}{2\times 3}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±22}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{4±22}{6}$$

Nā, me whakaoti te whārite x=\frac{4±22}{6} ina he tāpiri te ±. Tāpiri 4 ki te 22.

$$x=\frac{26}{6}$$

Whakahekea te hautanga \frac{26}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{13}{3}$$

Nā, me whakaoti te whārite x=\frac{4±22}{6} ina he tango te ±. Tango 22 mai i 4.

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

Whakawehe -18 ki te 6.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=\frac{13}{3} x=-3$$

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.

$$3x^{2}-4x=39$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-4x}{3}=\frac{39}{3}$$

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

$$x^{2}-\frac{4}{3}x=\frac{39}{3}$$

Whakawehe 39 ki te 3.

$$x^{2}-\frac{4}{3}x=13$$

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.

$$x^{2}-\frac{4}{3}x+\left(-\frac{2}{3}\right)^{2}=13+\left(-\frac{2}{3}\right)^{2}$$

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

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=13+\frac{4}{9}$$

Tāpiri 13 ki te \frac{4}{9}.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{121}{9}$$

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}.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{121}{9}$$

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

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{121}{9}}$$

Whakarūnātia.

$$x-\frac{2}{3}=\frac{11}{3} x-\frac{2}{3}=-\frac{11}{3}$$

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

$$x=\frac{13}{3} x=-3$$