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

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

Whakaoti mō x

$$x=\frac{\sqrt{111}-6}{5}\approx 0.907130751$$

$$x=\frac{-\sqrt{111}-6}{5}\approx -3.307130751$$

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.

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

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

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

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

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{3} mō a, \frac{4}{5} mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{4}{5}±\sqrt{\left(\frac{4}{5}\right)^{2}-4\times \frac{1}{3}\left(-1\right)}}{2\times \frac{1}{3}}$$

Pūruatia \frac{4}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{16}{25}-4\times \frac{1}{3}\left(-1\right)}}{2\times \frac{1}{3}}$$

Whakareatia -4 ki te \frac{1}{3}.

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{16}{25}-\frac{4}{3}\left(-1\right)}}{2\times \frac{1}{3}}$$

Whakareatia -\frac{4}{3} ki te -1.

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{16}{25}+\frac{4}{3}}}{2\times \frac{1}{3}}$$

Tāpiri \frac{16}{25} ki te \frac{4}{3} 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.

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{148}{75}}}{2\times \frac{1}{3}}$$

Tuhia te pūtakerua o te \frac{148}{75}.

$$x=\frac{-\frac{4}{5}±\frac{2\sqrt{111}}{15}}{2\times \frac{1}{3}}$$

Whakareatia 2 ki te \frac{1}{3}.

$$x=\frac{-\frac{4}{5}±\frac{2\sqrt{111}}{15}}{\frac{2}{3}}$$

Nā, me whakaoti te whārite x=\frac{-\frac{4}{5}±\frac{2\sqrt{111}}{15}}{\frac{2}{3}} ina he tāpiri te ±. Tāpiri -\frac{4}{5} ki te \frac{2\sqrt{111}}{15}.

$$x=\frac{\frac{2\sqrt{111}}{15}-\frac{4}{5}}{\frac{2}{3}}$$

Whakawehe -\frac{4}{5}+\frac{2\sqrt{111}}{15} ki te \frac{2}{3} mā te whakarea -\frac{4}{5}+\frac{2\sqrt{111}}{15} ki te tau huripoki o \frac{2}{3}.

$$x=\frac{\sqrt{111}-6}{5}$$

Nā, me whakaoti te whārite x=\frac{-\frac{4}{5}±\frac{2\sqrt{111}}{15}}{\frac{2}{3}} ina he tango te ±. Tango \frac{2\sqrt{111}}{15} mai i -\frac{4}{5}.

$$x=\frac{-\frac{2\sqrt{111}}{15}-\frac{4}{5}}{\frac{2}{3}}$$

Whakawehe -\frac{4}{5}-\frac{2\sqrt{111}}{15} ki te \frac{2}{3} mā te whakarea -\frac{4}{5}-\frac{2\sqrt{111}}{15} ki te tau huripoki o \frac{2}{3}.

$$x=\frac{-\sqrt{111}-6}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{111}-6}{5} x=\frac{-\sqrt{111}-6}{5}$$

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.

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

Me whakarea ngā taha e rua ki te 3.

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

Mā te whakawehe ki te \frac{1}{3} ka wetekia te whakareanga ki te \frac{1}{3}.

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

Whakawehe \frac{4}{5} ki te \frac{1}{3} mā te whakarea \frac{4}{5} ki te tau huripoki o \frac{1}{3}.

$$x^{2}+\frac{12}{5}x=\frac{1}{\frac{1}{3}}$$

Whakawehe 1 ki te \frac{1}{3} mā te whakarea 1 ki te tau huripoki o \frac{1}{3}.

$$x^{2}+\frac{12}{5}x=3$$

Whakawehea te \frac{12}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{6}{5}. Nā, tāpiria te pūrua o te \frac{6}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{12}{5}x+\left(\frac{6}{5}\right)^{2}=3+\left(\frac{6}{5}\right)^{2}$$

Pūruatia \frac{6}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=3+\frac{36}{25}$$

Tāpiri 3 ki te \frac{36}{25}.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=\frac{111}{25}$$

Tauwehea x^{2}+\frac{12}{5}x+\frac{36}{25}. 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}.

$$\left(x+\frac{6}{5}\right)^{2}=\frac{111}{25}$$

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

$$\sqrt{\left(x+\frac{6}{5}\right)^{2}}=\sqrt{\frac{111}{25}}$$

Whakarūnātia.

$$x+\frac{6}{5}=\frac{\sqrt{111}}{5} x+\frac{6}{5}=-\frac{\sqrt{111}}{5}$$

Me tango \frac{6}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{111}-6}{5} x=\frac{-\sqrt{111}-6}{5}$$