Whakaoti i te frac{sqrt{2}}{2}2+1/sqrt{3}=3x^2+15/sqrt{3}

$$\frac{ \sqrt{ 2 } }{ 2 } 2+ \frac{ 1 }{ \sqrt{ 3 } } = \frac{ 3 { x }^{ 2 } +15 }{ \sqrt{ 3 } }$$

Whakaoti mō x (complex solution)

$$x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}\approx -0-1.962185028i$$

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3}\approx 1.962185028i$$

Ngā Upane mā te Kimi Pūtakerua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}\left(3x^{2}+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{2}{3}\times 3^{\frac{1}{2}} ki te 3x^{2}+15.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}$$

Tangohia te 10\times 3^{\frac{1}{2}} mai i ngā taha e rua.

$$2\times 3^{\frac{1}{2}}x^{2}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}-10\times 3^{\frac{1}{2}}$$

Pahekotia te \frac{2}{3}\times 3^{\frac{1}{2}} me -10\times 3^{\frac{1}{2}}, ka -\frac{28}{3}\times 3^{\frac{1}{2}}.

$$2\times 3^{\frac{1}{2}}x^{2}=2\sqrt{2}-\frac{28}{3}\times 3^{\frac{1}{2}}$$

Whakaraupapatia anō ngā kīanga tau.

$$2\sqrt{3}x^{2}=-\frac{28}{3}\sqrt{3}+2\sqrt{2}$$

Mā te whakawehe ki te 2\sqrt{3} ka wetekia te whakareanga ki te 2\sqrt{3}.

$$x^{2}=\frac{-\frac{28\sqrt{3}}{3}+2\sqrt{2}}{2\sqrt{3}}$$

Whakawehe -\frac{28\sqrt{3}}{3}+2\sqrt{2} ki te 2\sqrt{3}.

$$x^{2}=\frac{\sqrt{6}-14}{3}$$

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

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3} x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=\frac{2}{3}\times 3^{\frac{1}{2}}\left(3x^{2}+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{2}{3}\times 3^{\frac{1}{2}} ki te 3x^{2}+15.

$$2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}=2\sqrt{2}+\frac{2}{3}\times 3^{\frac{1}{2}}$$

Tangohia te 2\sqrt{2} mai i ngā taha e rua.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}-2\sqrt{2}=\frac{2}{3}\times 3^{\frac{1}{2}}$$

Tangohia te \frac{2}{3}\times 3^{\frac{1}{2}} mai i ngā taha e rua.

$$2\times 3^{\frac{1}{2}}x^{2}+10\times 3^{\frac{1}{2}}-2\sqrt{2}-\frac{2}{3}\times 3^{\frac{1}{2}}=0$$

Pahekotia te 10\times 3^{\frac{1}{2}} me -\frac{2}{3}\times 3^{\frac{1}{2}}, ka \frac{28}{3}\times 3^{\frac{1}{2}}.

$$2\times 3^{\frac{1}{2}}x^{2}+\frac{28}{3}\times 3^{\frac{1}{2}}-2\sqrt{2}=0$$

Whakaraupapatia anō ngā kīanga tau.

$$2\sqrt{3}x^{2}-2\sqrt{2}+\frac{28}{3}\sqrt{3}=0$$

Ko ngā tikanga tātai pūrua pēnei i tēnei nā, me te kīanga tau x^{2} engari kāore he kīanga tau x, ka taea tonu te whakaoti mā te whakamahi i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, ina tuhia ki te tānga ngahuru: ax^{2}+bx+c=0.

$$2\sqrt{3}x^{2}+\frac{28\sqrt{3}}{3}-2\sqrt{2}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2\sqrt{3} mō a, 0 mō b, me -2\sqrt{2}+\frac{28\sqrt{3}}{3} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\sqrt{3}\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

Pūrua 0.

$$x=\frac{0±\sqrt{-4\times 2\sqrt{3}\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

Whakareatia -4 ki te 2\sqrt{3}.

$$x=\frac{0±\sqrt{\left(-8\sqrt{3}\right)\left(\frac{28\sqrt{3}}{3}-2\sqrt{2}\right)}}{2\times 2\sqrt{3}}$$

Whakareatia -8\sqrt{3} ki te -2\sqrt{2}+\frac{28\sqrt{3}}{3}.

$$x=\frac{0±\sqrt{16\sqrt{6}-224}}{2\times 2\sqrt{3}}$$

Tuhia te pūtakerua o te 16\sqrt{6}-224.

$$x=\frac{0±4i\sqrt{14-\sqrt{6}}}{2\times 2\sqrt{3}}$$

Whakareatia 2 ki te 2\sqrt{3}.

$$x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}}$$

Nā, me whakaoti te whārite x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}} ina he tāpiri te ±.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

Nā, me whakaoti te whārite x=\frac{0±4i\sqrt{14-\sqrt{6}}}{4\sqrt{3}} ina he tango te ±.

$$x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{i\sqrt{42-3\sqrt{6}}}{3} x=-\frac{i\sqrt{42-3\sqrt{6}}}{3}$$