Whakaoti i te 3x^2-2x-9=0

$$3 { x }^{ 2 } -2x-9 = 0$$

Whakaoti mō x

$$x = \frac{2 \sqrt{7} + 1}{3} \approx 2.097167541$$

$$x=\frac{1-2\sqrt{7}}{3}\approx -1.430500874$$

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.

$$3x^{2}-2x-9=0$$

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

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

Pūrua -2.

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

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-2\right)±\sqrt{4-12\left(-9\right)}}{2\times 3}$$

Whakareatia -12 ki te -9.

$$x=\frac{-\left(-2\right)±\sqrt{4+108}}{2\times 3}$$

Tāpiri 4 ki te 108.

$$x=\frac{-\left(-2\right)±\sqrt{112}}{2\times 3}$$

Tuhia te pūtakerua o te 112.

$$x=\frac{-\left(-2\right)±4\sqrt{7}}{2\times 3}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±4\sqrt{7}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{2±4\sqrt{7}}{6}$$

Nā, me whakaoti te whārite x=\frac{2±4\sqrt{7}}{6} ina he tāpiri te ±. Tāpiri 2 ki te 4\sqrt{7}.

$$x=\frac{4\sqrt{7}+2}{6}$$

Whakawehe 2+4\sqrt{7} ki te 6.

$$x=\frac{2\sqrt{7}+1}{3}$$

Nā, me whakaoti te whārite x=\frac{2±4\sqrt{7}}{6} ina he tango te ±. Tango 4\sqrt{7} mai i 2.

$$x=\frac{2-4\sqrt{7}}{6}$$

Whakawehe 2-4\sqrt{7} ki te 6.

$$x=\frac{1-2\sqrt{7}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{7}+1}{3} x=\frac{1-2\sqrt{7}}{3}$$

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.

$$3x^{2}-2x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

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

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

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

Tango -9 mai i 0.

$$3x^{2}-2x=9$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakawehe 9 ki te 3.

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

Whakawehea te -\frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{3}. Nā, tāpiria te pūrua o te -\frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

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

Tāpiri 3 ki te \frac{1}{9}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{28}{9}$$

Tauwehea x^{2}-\frac{2}{3}x+\frac{1}{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}.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{28}{9}$$

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

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{28}{9}}$$

Whakarūnātia.

$$x-\frac{1}{3}=\frac{2\sqrt{7}}{3} x-\frac{1}{3}=-\frac{2\sqrt{7}}{3}$$

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{7}+1}{3} x=\frac{1-2\sqrt{7}}{3}$$