Whakaoti i te 6x^2+18x-19=0

$$6 x ^ { 2 } + 18 x - 19 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2}\approx 0.827373341$$

$$x=-\frac{\sqrt{195}}{6}-\frac{3}{2}\approx -3.827373341$$

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.

$$6x^{2}+18x-19=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 18 mō b, me -19 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 6\left(-19\right)}}{2\times 6}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 6\left(-19\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-18±\sqrt{324-24\left(-19\right)}}{2\times 6}$$

Whakareatia -24 ki te -19.

$$x=\frac{-18±\sqrt{324+456}}{2\times 6}$$

Tāpiri 324 ki te 456.

$$x=\frac{-18±\sqrt{780}}{2\times 6}$$

Tuhia te pūtakerua o te 780.

$$x=\frac{-18±2\sqrt{195}}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-18±2\sqrt{195}}{12}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{195}}{12} ina he tāpiri te ±. Tāpiri -18 ki te 2\sqrt{195}.

$$x=\frac{2\sqrt{195}-18}{12}$$

Whakawehe -18+2\sqrt{195} ki te 12.

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{195}}{12} ina he tango te ±. Tango 2\sqrt{195} mai i -18.

$$x=\frac{-2\sqrt{195}-18}{12}$$

Whakawehe -18-2\sqrt{195} ki te 12.

$$x=-\frac{\sqrt{195}}{6}-\frac{3}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2} x=-\frac{\sqrt{195}}{6}-\frac{3}{2}$$

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.

$$6x^{2}+18x-19=0$$

Me tāpiri 19 ki ngā taha e rua o te whārite.

$$6x^{2}+18x-19-\left(-19\right)=-\left(-19\right)$$

Mā te tango i te -19 i a ia ake anō ka toe ko te 0.

$$6x^{2}+18x=-\left(-19\right)$$

Tango -19 mai i 0.

$$6x^{2}+18x=19$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+18x}{6}=\frac{19}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{18}{6}x=\frac{19}{6}$$

Whakawehe 18 ki te 6.

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

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

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

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

$$x^{2}+3x+\frac{9}{4}=\frac{19}{6}+\frac{9}{4}$$

Tāpiri \frac{19}{6} ki te \frac{9}{4} 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.

$$x^{2}+3x+\frac{9}{4}=\frac{65}{12}$$

Tauwehea x^{2}+3x+\frac{9}{4}. 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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{65}{12}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{65}{12}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{\sqrt{195}}{6} x+\frac{3}{2}=-\frac{\sqrt{195}}{6}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2} x=-\frac{\sqrt{195}}{6}-\frac{3}{2}$$