Whakaoti i te 3x^2+5x=2

$$3 x ^ { 2 } + 5 x = 2$$

Whakaoti mō x

$$x=-2$$

$$x=\frac{1}{3}\approx 0.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 2 mai i ngā taha e rua.

$$3x^{2}+5x-2=0$$

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

$$a+b=5 ab=3\left(-2\right)=-6$$

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 -6.

$$-1,6 -2,3$$

Tātaihia te tapeke mō ia takirua.

$$-1+6=5 -2+3=1$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-1 b=6$$

Tuhia anō te 3x^{2}+5x-2 hei \left(3x^{2}-x\right)+\left(6x-2\right).

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

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

$$x\left(3x-1\right)+2\left(3x-1\right)$$

Whakatauwehea atu te kīanga pātahi 3x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-1\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-1=0 me te x+2=0.

$$x=\frac{1}{3} x=-2$$

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.

$$3x^{2}+5x=2$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$3x^{2}+5x-2=2-2$$

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

$$3x^{2}+5x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 5 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\left(-2\right)}}{2\times 3}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 3\left(-2\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-5±\sqrt{25-12\left(-2\right)}}{2\times 3}$$

Whakareatia -12 ki te -2.

$$x=\frac{-5±\sqrt{25+24}}{2\times 3}$$

Tāpiri 25 ki te 24.

$$x=\frac{-5±\sqrt{49}}{2\times 3}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-5±7}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-5±7}{6}$$

Nā, me whakaoti te whārite x=\frac{-5±7}{6} ina he tāpiri te ±. Tāpiri -5 ki te 7.

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

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

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

Nā, me whakaoti te whārite x=\frac{-5±7}{6} ina he tango te ±. Tango 7 mai i -5.

$$x=-\frac{12}{6}$$

Whakawehe -12 ki te 6.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{3} x=-2$$

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.

$$3x^{2}+5x=2$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+5x}{3}=\frac{2}{3}$$

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

$$x^{2}+\frac{5}{3}x=\frac{2}{3}$$

Whakawehea te \frac{5}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{6}. Nā, tāpiria te pūrua o te \frac{5}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{2}{3}+\left(\frac{5}{6}\right)^{2}$$

Pūruatia \frac{5}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}$$

Tāpiri \frac{2}{3} ki te \frac{25}{36} 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.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{49}{36}$$

Tauwehea te x^{2}+\frac{5}{3}x+\frac{25}{36}. 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}.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{49}{36}$$

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

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Whakarūnātia.

$$x+\frac{5}{6}=\frac{7}{6} x+\frac{5}{6}=-\frac{7}{6}$$

Me tango \frac{5}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{3} x=-2$$