Whakaoti i te 14x^2+3x-2=0

$$14 x ^ { 2 } + 3 x - 2 = 0$$

Whakaoti mō x

$$x=-\frac{1}{2}=-0.5$$

$$x=\frac{2}{7}\approx 0.285714286$$

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 14x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=14\left(-2\right)=-28$$

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 -28.

$$-1,28 -2,14 -4,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+28=27 -2+14=12 -4+7=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-4 b=7$$

Tuhia anō te 14x^{2}+3x-2 hei \left(14x^{2}-4x\right)+\left(7x-2\right).

$$\left(14x^{2}-4x\right)+\left(7x-2\right)$$

Whakatauwehea atu 2x i te 14x^{2}-4x.

$$2x\left(7x-2\right)+7x-2$$

Whakatauwehea atu te kīanga pātahi 7x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-2\right)\left(2x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 7x-2=0 me te 2x+1=0.

$$x=\frac{2}{7} x=-\frac{1}{2}$$

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.

$$14x^{2}+3x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 14 mō a, 3 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 14\left(-2\right)}}{2\times 14}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\times 14\left(-2\right)}}{2\times 14}$$

Whakareatia -4 ki te 14.

$$x=\frac{-3±\sqrt{9-56\left(-2\right)}}{2\times 14}$$

Whakareatia -56 ki te -2.

$$x=\frac{-3±\sqrt{9+112}}{2\times 14}$$

Tāpiri 9 ki te 112.

$$x=\frac{-3±\sqrt{121}}{2\times 14}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-3±11}{2\times 14}$$

Whakareatia 2 ki te 14.

$$x=\frac{-3±11}{28}$$

Nā, me whakaoti te whārite x=\frac{-3±11}{28} ina he tāpiri te ±. Tāpiri -3 ki te 11.

$$x=\frac{8}{28}$$

Whakahekea te hautanga \frac{8}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{2}{7}$$

Nā, me whakaoti te whārite x=\frac{-3±11}{28} ina he tango te ±. Tango 11 mai i -3.

$$x=-\frac{14}{28}$$

Whakahekea te hautanga \frac{-14}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

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

Kua oti te whārite te whakatau.

$$x=\frac{2}{7} x=-\frac{1}{2}$$

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.

$$14x^{2}+3x-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$14x^{2}+3x-2-\left(-2\right)=-\left(-2\right)$$

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

$$14x^{2}+3x=-\left(-2\right)$$

Tango -2 mai i 0.

$$14x^{2}+3x=2$$

Whakawehea ngā taha e rua ki te 14.

$$\frac{14x^{2}+3x}{14}=\frac{2}{14}$$

Mā te whakawehe ki te 14 ka wetekia te whakareanga ki te 14.

$$x^{2}+\frac{3}{14}x=\frac{2}{14}$$

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

$$x^{2}+\frac{3}{14}x=\frac{1}{7}$$

Whakawehea te \frac{3}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{28}. Nā, tāpiria te pūrua o te \frac{3}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{14}x+\left(\frac{3}{28}\right)^{2}=\frac{1}{7}+\left(\frac{3}{28}\right)^{2}$$

Pūruatia \frac{3}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{14}x+\frac{9}{784}=\frac{1}{7}+\frac{9}{784}$$

Tāpiri \frac{1}{7} ki te \frac{9}{784} 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.

$$x^{2}+\frac{3}{14}x+\frac{9}{784}=\frac{121}{784}$$

Tauwehea te x^{2}+\frac{3}{14}x+\frac{9}{784}. 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}.

$$\left(x+\frac{3}{28}\right)^{2}=\frac{121}{784}$$

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

$$\sqrt{\left(x+\frac{3}{28}\right)^{2}}=\sqrt{\frac{121}{784}}$$

Whakarūnātia.

$$x+\frac{3}{28}=\frac{11}{28} x+\frac{3}{28}=-\frac{11}{28}$$

Me tango \frac{3}{28} mai i ngā taha e rua o te whārite.

$$x=\frac{2}{7} x=-\frac{1}{2}$$