Whakaoti i te 3x^2-19x-18=0

$$3 { x }^{ 2 } -19x-18=0$$

Whakaoti mō x

$$x = \frac{\sqrt{577} + 19}{6} \approx 7.170137383$$

$$x=\frac{19-\sqrt{577}}{6}\approx -0.83680405$$

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.

$$3x^{2}-19x-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -19 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 3\left(-18\right)}}{2\times 3}$$

Pūrua -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 3\left(-18\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-19\right)±\sqrt{361-12\left(-18\right)}}{2\times 3}$$

Whakareatia -12 ki te -18.

$$x=\frac{-\left(-19\right)±\sqrt{361+216}}{2\times 3}$$

Tāpiri 361 ki te 216.

$$x=\frac{-\left(-19\right)±\sqrt{577}}{2\times 3}$$

Ko te tauaro o -19 ko 19.

$$x=\frac{19±\sqrt{577}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{19±\sqrt{577}}{6}$$

Nā, me whakaoti te whārite x=\frac{19±\sqrt{577}}{6} ina he tāpiri te ±. Tāpiri 19 ki te \sqrt{577}.

$$x=\frac{\sqrt{577}+19}{6}$$

Nā, me whakaoti te whārite x=\frac{19±\sqrt{577}}{6} ina he tango te ±. Tango \sqrt{577} mai i 19.

$$x=\frac{19-\sqrt{577}}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{577}+19}{6} x=\frac{19-\sqrt{577}}{6}$$

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.

$$3x^{2}-19x-18=0$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$3x^{2}-19x-18-\left(-18\right)=-\left(-18\right)$$

Mā te tango i te -18 i a ia ake anō ka toe ko te 0.

$$3x^{2}-19x=-\left(-18\right)$$

Tango -18 mai i 0.

$$3x^{2}-19x=18$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-19x}{3}=\frac{18}{3}$$

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

$$x^{2}-\frac{19}{3}x=\frac{18}{3}$$

Whakawehe 18 ki te 3.

$$x^{2}-\frac{19}{3}x=6$$

Whakawehea te -\frac{19}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{6}. Nā, tāpiria te pūrua o te -\frac{19}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{19}{3}x+\left(-\frac{19}{6}\right)^{2}=6+\left(-\frac{19}{6}\right)^{2}$$

Pūruatia -\frac{19}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{19}{3}x+\frac{361}{36}=6+\frac{361}{36}$$

Tāpiri 6 ki te \frac{361}{36}.

$$x^{2}-\frac{19}{3}x+\frac{361}{36}=\frac{577}{36}$$

Tauwehea x^{2}-\frac{19}{3}x+\frac{361}{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}.

$$\left(x-\frac{19}{6}\right)^{2}=\frac{577}{36}$$

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

$$\sqrt{\left(x-\frac{19}{6}\right)^{2}}=\sqrt{\frac{577}{36}}$$

Whakarūnātia.

$$x-\frac{19}{6}=\frac{\sqrt{577}}{6} x-\frac{19}{6}=-\frac{\sqrt{577}}{6}$$

Me tāpiri \frac{19}{6} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{577}+19}{6} x=\frac{19-\sqrt{577}}{6}$$