Whakaoti i te 3x^2-4x-9=0

$$3 { x }^{ 2 } -4x-9=0$$

Whakaoti mō x

$$x = \frac{\sqrt{31} + 2}{3} \approx 2.522588121$$

$$x=\frac{2-\sqrt{31}}{3}\approx -1.189254788$$

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.

$$3x^{2}-4x-9=0$$

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

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

Pūrua -4.

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

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\left(-9\right)}}{2\times 3}$$

Whakareatia -12 ki te -9.

$$x=\frac{-\left(-4\right)±\sqrt{16+108}}{2\times 3}$$

Tāpiri 16 ki te 108.

$$x=\frac{-\left(-4\right)±\sqrt{124}}{2\times 3}$$

Tuhia te pūtakerua o te 124.

$$x=\frac{-\left(-4\right)±2\sqrt{31}}{2\times 3}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{31}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{4±2\sqrt{31}}{6}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{31}}{6} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{31}.

$$x=\frac{2\sqrt{31}+4}{6}$$

Whakawehe 4+2\sqrt{31} ki te 6.

$$x=\frac{\sqrt{31}+2}{3}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{31}}{6} ina he tango te ±. Tango 2\sqrt{31} mai i 4.

$$x=\frac{4-2\sqrt{31}}{6}$$

Whakawehe 4-2\sqrt{31} ki te 6.

$$x=\frac{2-\sqrt{31}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{31}+2}{3} x=\frac{2-\sqrt{31}}{3}$$

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.

$$3x^{2}-4x-9=0$$

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

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

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

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

Tango -9 mai i 0.

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

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakawehe 9 ki te 3.

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

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.

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

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

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

Tāpiri 3 ki te \frac{4}{9}.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{31}{9}$$

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}.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{31}{9}$$

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

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{31}{9}}$$

Whakarūnātia.

$$x-\frac{2}{3}=\frac{\sqrt{31}}{3} x-\frac{2}{3}=-\frac{\sqrt{31}}{3}$$

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

$$x=\frac{\sqrt{31}+2}{3} x=\frac{2-\sqrt{31}}{3}$$