Whakaoti i te left(3x+1right)^2=-2x

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

Whakaoti mō x

$$x=\frac{\sqrt{7}-4}{9}\approx -0.150472077$$

$$x=\frac{-\sqrt{7}-4}{9}\approx -0.738416812$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3x+1\right)^{2}.

$$9x^{2}+6x+1=-2x$$

Me tāpiri te 2x ki ngā taha e rua.

$$9x^{2}+6x+1+2x=0$$

Pahekotia te 6x me 2x, ka 8x.

$$9x^{2}+8x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 8 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 9}}{2\times 9}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 9}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-8±\sqrt{64-36}}{2\times 9}$$

Tāpiri 64 ki te -36.

$$x=\frac{-8±\sqrt{28}}{2\times 9}$$

Tuhia te pūtakerua o te 28.

$$x=\frac{-8±2\sqrt{7}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-8±2\sqrt{7}}{18}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{7}}{18} ina he tāpiri te ±. Tāpiri -8 ki te 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-8}{18}$$

Whakawehe -8+2\sqrt{7} ki te 18.

$$x=\frac{\sqrt{7}-4}{9}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{7}}{18} ina he tango te ±. Tango 2\sqrt{7} mai i -8.

$$x=\frac{-2\sqrt{7}-8}{18}$$

Whakawehe -8-2\sqrt{7} ki te 18.

$$x=\frac{-\sqrt{7}-4}{9}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{7}-4}{9} x=\frac{-\sqrt{7}-4}{9}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3x+1\right)^{2}.

$$9x^{2}+6x+1=-2x$$

Me tāpiri te 2x ki ngā taha e rua.

$$9x^{2}+6x+1+2x=0$$

Pahekotia te 6x me 2x, ka 8x.

$$9x^{2}+8x+1=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9x^{2}+8x=-1$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+8x}{9}=-\frac{1}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\frac{8}{9}x=-\frac{1}{9}$$

Whakawehea te \frac{8}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{4}{9}. Nā, tāpiria te pūrua o te \frac{4}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{8}{9}x+\left(\frac{4}{9}\right)^{2}=-\frac{1}{9}+\left(\frac{4}{9}\right)^{2}$$

Pūruatia \frac{4}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{8}{9}x+\frac{16}{81}=-\frac{1}{9}+\frac{16}{81}$$

Tāpiri -\frac{1}{9} ki te \frac{16}{81} 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.

$$x^{2}+\frac{8}{9}x+\frac{16}{81}=\frac{7}{81}$$

Tauwehea x^{2}+\frac{8}{9}x+\frac{16}{81}. 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}.

$$\left(x+\frac{4}{9}\right)^{2}=\frac{7}{81}$$

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

$$\sqrt{\left(x+\frac{4}{9}\right)^{2}}=\sqrt{\frac{7}{81}}$$

Whakarūnātia.

$$x+\frac{4}{9}=\frac{\sqrt{7}}{9} x+\frac{4}{9}=-\frac{\sqrt{7}}{9}$$

Me tango \frac{4}{9} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{7}-4}{9} x=\frac{-\sqrt{7}-4}{9}$$