Whakaoti i te (x)=3x^2-7x+2

$$( x ) = 3 x ^ { 2 } - 7 x + 2$$

Whakaoti mō x

$$x = \frac{\sqrt{10} + 4}{3} \approx 2.387425887$$

$$x=\frac{4-\sqrt{10}}{3}\approx 0.27924078$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3x^{2} mai i ngā taha e rua.

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

Me tāpiri te 7x ki ngā taha e rua.

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

Pahekotia te x me 7x, ka 8x.

$$8x-3x^{2}=2$$

Tangohia te 2 mai i ngā taha e rua.

$$8x-3x^{2}-2=0$$

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.

$$-3x^{2}+8x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 8 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-8±\sqrt{64+12\left(-2\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -2.

$$x=\frac{-8±\sqrt{64-24}}{2\left(-3\right)}$$

Tāpiri 64 ki te -24.

$$x=\frac{-8±\sqrt{40}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 40.

$$x=\frac{-8±2\sqrt{10}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-8±2\sqrt{10}}{-6}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{10}}{-6} ina he tāpiri te ±. Tāpiri -8 ki te 2\sqrt{10}.

$$x=\frac{2\sqrt{10}-8}{-6}$$

Whakawehe -8+2\sqrt{10} ki te -6.

$$x=\frac{4-\sqrt{10}}{3}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{10}}{-6} ina he tango te ±. Tango 2\sqrt{10} mai i -8.

$$x=\frac{-2\sqrt{10}-8}{-6}$$

Whakawehe -8-2\sqrt{10} ki te -6.

$$x=\frac{\sqrt{10}+4}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{4-\sqrt{10}}{3} x=\frac{\sqrt{10}+4}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3x^{2} mai i ngā taha e rua.

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

Me tāpiri te 7x ki ngā taha e rua.

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

Pahekotia te x me 7x, ka 8x.

$$8x-3x^{2}=2$$

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.

$$-3x^{2}+8x=2$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+8x}{-3}=\frac{2}{-3}$$

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

$$x^{2}+\frac{8}{-3}x=\frac{2}{-3}$$

Whakawehe 8 ki te -3.

$$x^{2}-\frac{8}{3}x=\frac{2}{-3}$$

Whakawehe 2 ki te -3.

$$x^{2}-\frac{8}{3}x=-\frac{2}{3}$$

Whakawehea te -\frac{8}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4}{3}. Nā, tāpiria te pūrua o te -\frac{4}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=-\frac{2}{3}+\left(-\frac{4}{3}\right)^{2}$$

Pūruatia -\frac{4}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=-\frac{2}{3}+\frac{16}{9}$$

Tāpiri -\frac{2}{3} ki te \frac{16}{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.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{10}{9}$$

Tauwehea x^{2}-\frac{8}{3}x+\frac{16}{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}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{10}{9}$$

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

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{10}{9}}$$

Whakarūnātia.

$$x-\frac{4}{3}=\frac{\sqrt{10}}{3} x-\frac{4}{3}=-\frac{\sqrt{10}}{3}$$

Me tāpiri \frac{4}{3} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{10}+4}{3} x=\frac{4-\sqrt{10}}{3}$$