Whakaoti i te {l}{x-y=1/4}{3x^2-6=y^2}

$$\left\{ \begin{array} { l } { x - y = \frac { 1 } { 4 } } \\ { 3 x ^ { 2 } - 6 = y ^ { 2 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{-\sqrt{195}-1}{8}\approx -1.870530005\text{, }y=\frac{-\sqrt{195}-3}{8}\approx -2.120530005$$

$$x=\frac{\sqrt{195}-1}{8}\approx 1.620530005\text{, }y=\frac{\sqrt{195}-3}{8}\approx 1.370530005$$

Ngā Upane e Whakamahi ana i te Whakakapinga me te Tikanga Tātai Pūrua

Whakaarohia te whārite tuarua. Tangohia te y^{2} mai i ngā taha e rua.

$$3x^{2}-6-y^{2}=0$$

Me tāpiri te 6 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x^{2}-y^{2}=6$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$x-y=\frac{1}{4},-y^{2}+3x^{2}=6$$

Whakaotia te x-y=\frac{1}{4} mō x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$x-y=\frac{1}{4}$$

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

$$x=y+\frac{1}{4}$$

Whakakapia te y+\frac{1}{4} mō te x ki tērā atu whārite, -y^{2}+3x^{2}=6.

$$-y^{2}+3\left(y+\frac{1}{4}\right)^{2}=6$$

Pūrua y+\frac{1}{4}.

$$-y^{2}+3\left(y^{2}+\frac{1}{2}y+\frac{1}{16}\right)=6$$

Whakareatia 3 ki te y^{2}+\frac{1}{2}y+\frac{1}{16}.

$$-y^{2}+3y^{2}+\frac{3}{2}y+\frac{3}{16}=6$$

Tāpiri -y^{2} ki te 3y^{2}.

$$2y^{2}+\frac{3}{2}y+\frac{3}{16}=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$2y^{2}+\frac{3}{2}y-\frac{93}{16}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1+3\times 1^{2} mō a, 3\times \frac{1}{4}\times 1\times 2 mō b, me -\frac{93}{16} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{3}{2}±\sqrt{\left(\frac{3}{2}\right)^{2}-4\times 2\left(-\frac{93}{16}\right)}}{2\times 2}$$

Pūrua 3\times \frac{1}{4}\times 1\times 2.

$$y=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}-4\times 2\left(-\frac{93}{16}\right)}}{2\times 2}$$

Whakareatia -4 ki te -1+3\times 1^{2}.

$$y=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}-8\left(-\frac{93}{16}\right)}}{2\times 2}$$

Whakareatia -8 ki te -\frac{93}{16}.

$$y=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}+\frac{93}{2}}}{2\times 2}$$

Tāpiri \frac{9}{4} ki te \frac{93}{2} 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.

$$y=\frac{-\frac{3}{2}±\sqrt{\frac{195}{4}}}{2\times 2}$$

Tuhia te pūtakerua o te \frac{195}{4}.

$$y=\frac{-\frac{3}{2}±\frac{\sqrt{195}}{2}}{2\times 2}$$

Whakareatia 2 ki te -1+3\times 1^{2}.

$$y=\frac{-\frac{3}{2}±\frac{\sqrt{195}}{2}}{4}$$

Nā, me whakaoti te whārite y=\frac{-\frac{3}{2}±\frac{\sqrt{195}}{2}}{4} ina he tāpiri te ±. Tāpiri -\frac{3}{2} ki te \frac{\sqrt{195}}{2}.

$$y=\frac{\sqrt{195}-3}{2\times 4}$$

Whakawehe \frac{-3+\sqrt{195}}{2} ki te 4.

$$y=\frac{\sqrt{195}-3}{8}$$

Nā, me whakaoti te whārite y=\frac{-\frac{3}{2}±\frac{\sqrt{195}}{2}}{4} ina he tango te ±. Tango \frac{\sqrt{195}}{2} mai i -\frac{3}{2}.

$$y=\frac{-\sqrt{195}-3}{2\times 4}$$

Whakawehe \frac{-3-\sqrt{195}}{2} ki te 4.

$$y=\frac{-\sqrt{195}-3}{8}$$

E rua ngā otinga mō y: \frac{-3+\sqrt{195}}{8} me \frac{-3-\sqrt{195}}{8}. Me whakakapi \frac{-3+\sqrt{195}}{8} mō y ki te whārite x=y+\frac{1}{4} hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=\frac{\sqrt{195}-3}{8}+\frac{1}{4}$$

Me whakakapi te \frac{-3-\sqrt{195}}{8} ināianei mō te y ki te whārite x=y+\frac{1}{4} ka whakaoti hei kimi i te otinga hāngai mō x e pai ai ki ngā whārite e rua.

$$x=\frac{-\sqrt{195}-3}{8}+\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{\sqrt{195}-3}{8}+\frac{1}{4},y=\frac{\sqrt{195}-3}{8}\text{ or }x=\frac{-\sqrt{195}-3}{8}+\frac{1}{4},y=\frac{-\sqrt{195}-3}{8}$$