Whakaoti i te 3x^2-4x-16=0

$$3 x ^ { 2 } - 4 x - 16 = 0$$

Whakaoti mō x

$$x = \frac{2 \sqrt{13} + 2}{3} \approx 3.070367517$$

$$x=\frac{2-2\sqrt{13}}{3}\approx -1.737034184$$

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.

$$3x^{2}-4x-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -4 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -4.

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

Whakareatia -4 ki te 3.

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

Whakareatia -12 ki te -16.

$$x=\frac{-\left(-4\right)±\sqrt{16+192}}{2\times 3}$$

Tāpiri 16 ki te 192.

$$x=\frac{-\left(-4\right)±\sqrt{208}}{2\times 3}$$

Tuhia te pūtakerua o te 208.

$$x=\frac{-\left(-4\right)±4\sqrt{13}}{2\times 3}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±4\sqrt{13}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{4±4\sqrt{13}}{6}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{13}}{6} ina he tāpiri te ±. Tāpiri 4 ki te 4\sqrt{13}.

$$x=\frac{4\sqrt{13}+4}{6}$$

Whakawehe 4+4\sqrt{13} ki te 6.

$$x=\frac{2\sqrt{13}+2}{3}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{13}}{6} ina he tango te ±. Tango 4\sqrt{13} mai i 4.

$$x=\frac{4-4\sqrt{13}}{6}$$

Whakawehe 4-4\sqrt{13} ki te 6.

$$x=\frac{2-2\sqrt{13}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{13}+2}{3} x=\frac{2-2\sqrt{13}}{3}$$

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.

$$3x^{2}-4x-16=0$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$3x^{2}-4x-16-\left(-16\right)=-\left(-16\right)$$

Mā te tango i te -16 i a ia ake anō ka toe ko te 0.

$$3x^{2}-4x=-\left(-16\right)$$

Tango -16 mai i 0.

$$3x^{2}-4x=16$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-4x}{3}=\frac{16}{3}$$

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

$$x^{2}-\frac{4}{3}x=\frac{16}{3}$$

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.

$$x^{2}-\frac{4}{3}x+\left(-\frac{2}{3}\right)^{2}=\frac{16}{3}+\left(-\frac{2}{3}\right)^{2}$$

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

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{16}{3}+\frac{4}{9}$$

Tāpiri \frac{16}{3} ki te \frac{4}{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.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{52}{9}$$

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}.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{52}{9}$$

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

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{52}{9}}$$

Whakarūnātia.

$$x-\frac{2}{3}=\frac{2\sqrt{13}}{3} x-\frac{2}{3}=-\frac{2\sqrt{13}}{3}$$

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

$$x=\frac{2\sqrt{13}+2}{3} x=\frac{2-2\sqrt{13}}{3}$$