Whakaoti i te 3x(x+1)=215

$$3x(x+1)=215$$

Whakaoti mō x

$$x=\frac{\sqrt{2589}}{6}-\frac{1}{2}\approx 7.980369489$$

$$x=-\frac{\sqrt{2589}}{6}-\frac{1}{2}\approx -8.980369489$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te x+1.

$$3x^{2}+3x=215$$

Tangohia te 215 mai i ngā taha e rua.

$$3x^{2}+3x-215=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 3 mō b, me -215 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 3\left(-215\right)}}{2\times 3}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\times 3\left(-215\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-3±\sqrt{9-12\left(-215\right)}}{2\times 3}$$

Whakareatia -12 ki te -215.

$$x=\frac{-3±\sqrt{9+2580}}{2\times 3}$$

Tāpiri 9 ki te 2580.

$$x=\frac{-3±\sqrt{2589}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-3±\sqrt{2589}}{6}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{2589}}{6} ina he tāpiri te ±. Tāpiri -3 ki te \sqrt{2589}.

$$x=\frac{\sqrt{2589}-3}{6}$$

Whakawehe -3+\sqrt{2589} ki te 6.

$$x=\frac{\sqrt{2589}}{6}-\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{2589}}{6} ina he tango te ±. Tango \sqrt{2589} mai i -3.

$$x=\frac{-\sqrt{2589}-3}{6}$$

Whakawehe -3-\sqrt{2589} ki te 6.

$$x=-\frac{\sqrt{2589}}{6}-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{2589}}{6}-\frac{1}{2} x=-\frac{\sqrt{2589}}{6}-\frac{1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te x+1.

$$3x^{2}+3x=215$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+3x}{3}=\frac{215}{3}$$

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

$$x^{2}+\frac{3}{3}x=\frac{215}{3}$$

Whakawehe 3 ki te 3.

$$x^{2}+x=\frac{215}{3}$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{215}{3}+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=\frac{215}{3}+\frac{1}{4}$$

Tāpiri \frac{215}{3} ki te \frac{1}{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.

$$x^{2}+x+\frac{1}{4}=\frac{863}{12}$$

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

$$\left(x+\frac{1}{2}\right)^{2}=\frac{863}{12}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{863}{12}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{\sqrt{2589}}{6} x+\frac{1}{2}=-\frac{\sqrt{2589}}{6}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{2589}}{6}-\frac{1}{2} x=-\frac{\sqrt{2589}}{6}-\frac{1}{2}$$