Whakaoti i te 9x^2-4x-2=0

$$9 x ^ { 2 } - 4 x - 2 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{22}+2}{9}\approx 0.743379529$$

$$x=\frac{2-\sqrt{22}}{9}\approx -0.298935084$$

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.

$$9x^{2}-4x-2=0$$

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

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

Pūrua -4.

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

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-4\right)±\sqrt{16-36\left(-2\right)}}{2\times 9}$$

Whakareatia -36 ki te -2.

$$x=\frac{-\left(-4\right)±\sqrt{16+72}}{2\times 9}$$

Tāpiri 16 ki te 72.

$$x=\frac{-\left(-4\right)±\sqrt{88}}{2\times 9}$$

Tuhia te pūtakerua o te 88.

$$x=\frac{-\left(-4\right)±2\sqrt{22}}{2\times 9}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{22}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{4±2\sqrt{22}}{18}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{22}}{18} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{22}.

$$x=\frac{2\sqrt{22}+4}{18}$$

Whakawehe 4+2\sqrt{22} ki te 18.

$$x=\frac{\sqrt{22}+2}{9}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{22}}{18} ina he tango te ±. Tango 2\sqrt{22} mai i 4.

$$x=\frac{4-2\sqrt{22}}{18}$$

Whakawehe 4-2\sqrt{22} ki te 18.

$$x=\frac{2-\sqrt{22}}{9}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{22}+2}{9} x=\frac{2-\sqrt{22}}{9}$$

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.

$$9x^{2}-4x-2=0$$

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

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

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

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

Tango -2 mai i 0.

$$9x^{2}-4x=2$$

Whakawehea ngā taha e rua ki te 9.

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

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

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

Whakawehea te -\frac{4}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{9}. Nā, tāpiria te pūrua o te -\frac{2}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{2}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=\frac{2}{9}+\frac{4}{81}$$

Tāpiri \frac{2}{9} ki te \frac{4}{81} 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.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=\frac{22}{81}$$

Tauwehea x^{2}-\frac{4}{9}x+\frac{4}{81}. 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}.

$$\left(x-\frac{2}{9}\right)^{2}=\frac{22}{81}$$

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

$$\sqrt{\left(x-\frac{2}{9}\right)^{2}}=\sqrt{\frac{22}{81}}$$

Whakarūnātia.

$$x-\frac{2}{9}=\frac{\sqrt{22}}{9} x-\frac{2}{9}=-\frac{\sqrt{22}}{9}$$

Me tāpiri \frac{2}{9} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{22}+2}{9} x=\frac{2-\sqrt{22}}{9}$$