Whakaoti i te 3m^2+4m+1=5/9

$$3 m ^ { 2 } + 4 m + 1 = \frac { 5 } { 9 }$$

Whakaoti mō m

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3}\approx -0.122335613$$

$$m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}\approx -1.210997721$$

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.

$$3m^{2}+4m+1=\frac{5}{9}$$

Me tango \frac{5}{9} mai i ngā taha e rua o te whārite.

$$3m^{2}+4m+1-\frac{5}{9}=\frac{5}{9}-\frac{5}{9}$$

Mā te tango i te \frac{5}{9} i a ia ake anō ka toe ko te 0.

$$3m^{2}+4m+1-\frac{5}{9}=0$$

Tango \frac{5}{9} mai i 1.

$$3m^{2}+4m+\frac{4}{9}=0$$

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

$$m=\frac{-4±\sqrt{4^{2}-4\times 3\times \frac{4}{9}}}{2\times 3}$$

Pūrua 4.

$$m=\frac{-4±\sqrt{16-4\times 3\times \frac{4}{9}}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$m=\frac{-4±\sqrt{16-12\times \frac{4}{9}}}{2\times 3}$$

Whakareatia -12 ki te \frac{4}{9}.

$$m=\frac{-4±\sqrt{16-\frac{16}{3}}}{2\times 3}$$

Tāpiri 16 ki te -\frac{16}{3}.

$$m=\frac{-4±\sqrt{\frac{32}{3}}}{2\times 3}$$

Tuhia te pūtakerua o te \frac{32}{3}.

$$m=\frac{-4±\frac{4\sqrt{6}}{3}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$m=\frac{-4±\frac{4\sqrt{6}}{3}}{6}$$

Nā, me whakaoti te whārite m=\frac{-4±\frac{4\sqrt{6}}{3}}{6} ina he tāpiri te ±. Tāpiri -4 ki te \frac{4\sqrt{6}}{3}.

$$m=\frac{\frac{4\sqrt{6}}{3}-4}{6}$$

Whakawehe -4+\frac{4\sqrt{6}}{3} ki te 6.

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3}$$

Nā, me whakaoti te whārite m=\frac{-4±\frac{4\sqrt{6}}{3}}{6} ina he tango te ±. Tango \frac{4\sqrt{6}}{3} mai i -4.

$$m=\frac{-\frac{4\sqrt{6}}{3}-4}{6}$$

Whakawehe -4-\frac{4\sqrt{6}}{3} ki te 6.

$$m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}$$

Kua oti te whārite te whakatau.

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3} m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}$$

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.

$$3m^{2}+4m+1=\frac{5}{9}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$3m^{2}+4m+1-1=\frac{5}{9}-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$3m^{2}+4m=\frac{5}{9}-1$$

Tango 1 mai i \frac{5}{9}.

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

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakawehe -\frac{4}{9} ki te 3.

$$m^{2}+\frac{4}{3}m=-\frac{4}{27}$$

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.

$$m^{2}+\frac{4}{3}m+\left(\frac{2}{3}\right)^{2}=-\frac{4}{27}+\left(\frac{2}{3}\right)^{2}$$

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

$$m^{2}+\frac{4}{3}m+\frac{4}{9}=-\frac{4}{27}+\frac{4}{9}$$

Tāpiri -\frac{4}{27} ki te \frac{4}{9} 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.

$$m^{2}+\frac{4}{3}m+\frac{4}{9}=\frac{8}{27}$$

Tauwehea m^{2}+\frac{4}{3}m+\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}.

$$\left(m+\frac{2}{3}\right)^{2}=\frac{8}{27}$$

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

$$\sqrt{\left(m+\frac{2}{3}\right)^{2}}=\sqrt{\frac{8}{27}}$$

Whakarūnātia.

$$m+\frac{2}{3}=\frac{2\sqrt{6}}{9} m+\frac{2}{3}=-\frac{2\sqrt{6}}{9}$$

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

$$m=\frac{2\sqrt{6}}{9}-\frac{2}{3} m=-\frac{2\sqrt{6}}{9}-\frac{2}{3}$$