Whakaoti i te 1/3x^2-4/3x-4=16

$$\frac { 1 } { 3 } x ^ { 2 } - \frac { 4 } { 3 } x - 4 = 16$$

Whakaoti mō x

$$x=-6$$

$$x=10$$

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.

$$\frac{1}{3}x^{2}-\frac{4}{3}x-4=16$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$\frac{1}{3}x^{2}-\frac{4}{3}x-4-16=16-16$$

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

$$\frac{1}{3}x^{2}-\frac{4}{3}x-4-16=0$$

Tango 16 mai i -4.

$$\frac{1}{3}x^{2}-\frac{4}{3}x-20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{3} mō a, -\frac{4}{3} mō b, me -20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\left(-\frac{4}{3}\right)^{2}-4\times \frac{1}{3}\left(-20\right)}}{2\times \frac{1}{3}}$$

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

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}-4\times \frac{1}{3}\left(-20\right)}}{2\times \frac{1}{3}}$$

Whakareatia -4 ki te \frac{1}{3}.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}-\frac{4}{3}\left(-20\right)}}{2\times \frac{1}{3}}$$

Whakareatia -\frac{4}{3} ki te -20.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}+\frac{80}{3}}}{2\times \frac{1}{3}}$$

Tāpiri \frac{16}{9} ki te \frac{80}{3} 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.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{256}{9}}}{2\times \frac{1}{3}}$$

Tuhia te pūtakerua o te \frac{256}{9}.

$$x=\frac{-\left(-\frac{4}{3}\right)±\frac{16}{3}}{2\times \frac{1}{3}}$$

Ko te tauaro o -\frac{4}{3} ko \frac{4}{3}.

$$x=\frac{\frac{4}{3}±\frac{16}{3}}{2\times \frac{1}{3}}$$

Whakareatia 2 ki te \frac{1}{3}.

$$x=\frac{\frac{4}{3}±\frac{16}{3}}{\frac{2}{3}}$$

Nā, me whakaoti te whārite x=\frac{\frac{4}{3}±\frac{16}{3}}{\frac{2}{3}} ina he tāpiri te ±. Tāpiri \frac{4}{3} ki te \frac{16}{3} 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.

$$x=\frac{\frac{20}{3}}{\frac{2}{3}}$$

Whakawehe \frac{20}{3} ki te \frac{2}{3} mā te whakarea \frac{20}{3} ki te tau huripoki o \frac{2}{3}.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{\frac{4}{3}±\frac{16}{3}}{\frac{2}{3}} ina he tango te ±. Tango \frac{16}{3} mai i \frac{4}{3} mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Whakawehe -4 ki te \frac{2}{3} mā te whakarea -4 ki te tau huripoki o \frac{2}{3}.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=10 x=-6$$

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.

$$\frac{1}{3}x^{2}-\frac{4}{3}x-4=16$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$\frac{1}{3}x^{2}-\frac{4}{3}x-4-\left(-4\right)=16-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$\frac{1}{3}x^{2}-\frac{4}{3}x=16-\left(-4\right)$$

Tango -4 mai i 16.

$$\frac{1}{3}x^{2}-\frac{4}{3}x=20$$

Me whakarea ngā taha e rua ki te 3.

$$\frac{\frac{1}{3}x^{2}-\frac{4}{3}x}{\frac{1}{3}}=\frac{20}{\frac{1}{3}}$$

Mā te whakawehe ki te \frac{1}{3} ka wetekia te whakareanga ki te \frac{1}{3}.

$$x^{2}+\left(-\frac{\frac{4}{3}}{\frac{1}{3}}\right)x=\frac{20}{\frac{1}{3}}$$

Whakawehe -\frac{4}{3} ki te \frac{1}{3} mā te whakarea -\frac{4}{3} ki te tau huripoki o \frac{1}{3}.

$$x^{2}-4x=\frac{20}{\frac{1}{3}}$$

Whakawehe 20 ki te \frac{1}{3} mā te whakarea 20 ki te tau huripoki o \frac{1}{3}.

$$x^{2}-4x=60$$

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

$$x^{2}-4x+\left(-2\right)^{2}=60+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=60+4$$

Tāpiri 60 ki te 4.

$$x^{2}-4x+4=64$$

Tauwehea x^{2}-4x+4. 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}.

$$\left(x-2\right)^{2}=64$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$x-2=8 x-2=-8$$

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

$$x=10 x=-6$$