Whakaoti i te x^2/9-4/3x=-2

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

Whakaoti mō x

$$x=3\sqrt{2}+6\approx 10.242640687$$

$$x=6-3\sqrt{2}\approx 1.757359313$$

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.

$$\frac{1}{9}x^{2}-\frac{4}{3}x=-2$$

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

$$\frac{1}{9}x^{2}-\frac{4}{3}x-\left(-2\right)=-2-\left(-2\right)$$

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

$$\frac{1}{9}x^{2}-\frac{4}{3}x-\left(-2\right)=0$$

Tango -2 mai i 0.

$$\frac{1}{9}x^{2}-\frac{4}{3}x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{9} mō a, -\frac{4}{3} mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

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

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

Whakareatia -4 ki te \frac{1}{9}.

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

Whakareatia -\frac{4}{9} ki te 2.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16-8}{9}}}{2\times \frac{1}{9}}$$

Tāpiri \frac{16}{9} ki te -\frac{8}{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.

$$x=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{8}{9}}}{2\times \frac{1}{9}}$$

Tuhia te pūtakerua o te \frac{8}{9}.

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

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

$$x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{2\times \frac{1}{9}}$$

Whakareatia 2 ki te \frac{1}{9}.

$$x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{\frac{2}{9}}$$

Nā, me whakaoti te whārite x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{\frac{2}{9}} ina he tāpiri te ±. Tāpiri \frac{4}{3} ki te \frac{2\sqrt{2}}{3}.

$$x=\frac{2\sqrt{2}+4}{\frac{2}{9}\times 3}$$

Whakawehe \frac{4+2\sqrt{2}}{3} ki te \frac{2}{9} mā te whakarea \frac{4+2\sqrt{2}}{3} ki te tau huripoki o \frac{2}{9}.

$$x=3\sqrt{2}+6$$

Nā, me whakaoti te whārite x=\frac{\frac{4}{3}±\frac{2\sqrt{2}}{3}}{\frac{2}{9}} ina he tango te ±. Tango \frac{2\sqrt{2}}{3} mai i \frac{4}{3}.

$$x=\frac{4-2\sqrt{2}}{\frac{2}{9}\times 3}$$

Whakawehe \frac{4-2\sqrt{2}}{3} ki te \frac{2}{9} mā te whakarea \frac{4-2\sqrt{2}}{3} ki te tau huripoki o \frac{2}{9}.

$$x=6-3\sqrt{2}$$

Kua oti te whārite te whakatau.

$$x=3\sqrt{2}+6 x=6-3\sqrt{2}$$

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.

$$\frac{1}{9}x^{2}-\frac{4}{3}x=-2$$

Me whakarea ngā taha e rua ki te 9.

$$\frac{\frac{1}{9}x^{2}-\frac{4}{3}x}{\frac{1}{9}}=-\frac{2}{\frac{1}{9}}$$

Mā te whakawehe ki te \frac{1}{9} ka wetekia te whakareanga ki te \frac{1}{9}.

$$x^{2}+\left(-\frac{\frac{4}{3}}{\frac{1}{9}}\right)x=-\frac{2}{\frac{1}{9}}$$

Whakawehe -\frac{4}{3} ki te \frac{1}{9} mā te whakarea -\frac{4}{3} ki te tau huripoki o \frac{1}{9}.

$$x^{2}-12x=-\frac{2}{\frac{1}{9}}$$

Whakawehe -2 ki te \frac{1}{9} mā te whakarea -2 ki te tau huripoki o \frac{1}{9}.

$$x^{2}-12x=-18$$

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=-18+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=-18+36$$

Tāpiri -18 ki te 36.

$$x^{2}-12x+36=18$$

Tauwehea x^{2}-12x+36. 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}.

$$\left(x-6\right)^{2}=18$$

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

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{18}$$

Whakarūnātia.

$$x-6=3\sqrt{2} x-6=-3\sqrt{2}$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=3\sqrt{2}+6 x=6-3\sqrt{2}$$