Whakaoti i te 6x^2+4x-3=1

$$6 x ^ { 2 } + 4 x - 3 = 1$$

Whakaoti mō x

$$x=\frac{\sqrt{7}-1}{3}\approx 0.54858377$$

$$x=\frac{-\sqrt{7}-1}{3}\approx -1.215250437$$

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.

$$6x^{2}+4x-3=1$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$6x^{2}+4x-3-1=1-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$6x^{2}+4x-3-1=0$$

Tango 1 mai i -3.

$$6x^{2}+4x-4=0$$

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

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

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 6\left(-4\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-4±\sqrt{16-24\left(-4\right)}}{2\times 6}$$

Whakareatia -24 ki te -4.

$$x=\frac{-4±\sqrt{16+96}}{2\times 6}$$

Tāpiri 16 ki te 96.

$$x=\frac{-4±\sqrt{112}}{2\times 6}$$

Tuhia te pūtakerua o te 112.

$$x=\frac{-4±4\sqrt{7}}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-4±4\sqrt{7}}{12}$$

Nā, me whakaoti te whārite x=\frac{-4±4\sqrt{7}}{12} ina he tāpiri te ±. Tāpiri -4 ki te 4\sqrt{7}.

$$x=\frac{4\sqrt{7}-4}{12}$$

Whakawehe -4+4\sqrt{7} ki te 12.

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

Nā, me whakaoti te whārite x=\frac{-4±4\sqrt{7}}{12} ina he tango te ±. Tango 4\sqrt{7} mai i -4.

$$x=\frac{-4\sqrt{7}-4}{12}$$

Whakawehe -4-4\sqrt{7} ki te 12.

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

Kua oti te whārite te whakatau.

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

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.

$$6x^{2}+4x-3=1$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$6x^{2}+4x-3-\left(-3\right)=1-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$6x^{2}+4x=1-\left(-3\right)$$

Tango -3 mai i 1.

$$6x^{2}+4x=4$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+4x}{6}=\frac{4}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{4}{6}x=\frac{4}{6}$$

Whakahekea te hautanga \frac{4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{2}{3}x=\frac{4}{6}$$

Whakahekea te hautanga \frac{4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Whakawehea te \frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{3}. Nā, tāpiria te pūrua o te \frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia \frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

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

Tāpiri \frac{2}{3} ki te \frac{1}{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.

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

Tauwehea x^{2}+\frac{2}{3}x+\frac{1}{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}.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{7}{9}$$

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

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{7}{9}}$$

Whakarūnātia.

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

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

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