Whakaoti i te x=1/x+1/6

$$x = \frac { 1 } { x } + \frac { 1 } { 6 }$$

Whakaoti mō x

$$x = \frac{\sqrt{145} + 1}{12} \approx 1.086799548$$

$$x=\frac{1-\sqrt{145}}{12}\approx -0.920132882$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o x me 6 ko 6x. Whakareatia \frac{1}{x} ki te \frac{6}{6}. Whakareatia \frac{1}{6} ki te \frac{x}{x}.

$$x=\frac{6}{6x}+\frac{x}{6x}$$

Tā te mea he rite te tauraro o \frac{6}{6x} me \frac{x}{6x}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$x=\frac{6+x}{6x}$$

Tangohia te \frac{6+x}{6x} mai i ngā taha e rua.

$$x-\frac{6+x}{6x}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{6x}{6x}.

$$\frac{x\times 6x}{6x}-\frac{6+x}{6x}=0$$

Tā te mea he rite te tauraro o \frac{x\times 6x}{6x} me \frac{6+x}{6x}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x\times 6x-\left(6+x\right)}{6x}=0$$

Mahia ngā whakarea i roto o x\times 6x-\left(6+x\right).

$$\frac{6x^{2}-6-x}{6x}=0$$

Me whakatauwehe ngā kīanga kāore anō i whakatauwehea i roto o \frac{6x^{2}-6-x}{6x}.

$$\frac{6\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{6x}=0$$

Me whakakore tahi te 6 i te taurunga me te tauraro.

$$\frac{\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

Hei kimi i te tauaro o -\frac{1}{12}\sqrt{145}+\frac{1}{12}, kimihia te tauaro o ia taurangi.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}\right)-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

Ko te tauaro o -\frac{1}{12}\sqrt{145} ko \frac{1}{12}\sqrt{145}.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

Hei kimi i te tauaro o \frac{1}{12}\sqrt{145}+\frac{1}{12}, kimihia te tauaro o ia taurangi.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)=0$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o x+\frac{1}{12}\sqrt{145}-\frac{1}{12} ki ia tau o x-\frac{1}{12}\sqrt{145}-\frac{1}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)\sqrt{145}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Whakareatia te \sqrt{145} ki te \sqrt{145}, ka 145.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Pahekotia te x\left(-\frac{1}{12}\right)\sqrt{145} me \frac{1}{12}\sqrt{145}x, ka 0.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Whakareatia te \frac{1}{12} ki te 145, ka \frac{145}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145}{12}\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te \frac{145}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145\left(-1\right)}{12\times 12}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Mahia ngā whakarea i roto i te hautanga \frac{145\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{-145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Ka taea te hautanga \frac{-145}{144} te tuhi anō ko -\frac{145}{144} mā te tango i te tohu tōraro.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te \frac{1}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Mahia ngā whakarea i roto i te hautanga \frac{1\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{-1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Ka taea te hautanga \frac{-1}{144} te tuhi anō ko -\frac{1}{144} mā te tango i te tohu tōraro.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Pahekotia te x\left(-\frac{1}{12}\right) me -\frac{1}{12}x, ka -\frac{1}{6}x.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te -\frac{1}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{-\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Mahia ngā whakarea i roto i te hautanga \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Pahekotia te -\frac{1}{144}\sqrt{145} me \frac{1}{144}\sqrt{145}, ka 0.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te -\frac{1}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{-\left(-1\right)}{12\times 12}=0$$

Mahia ngā whakarea i roto i te hautanga \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{1}{144}=0$$

Tā te mea he rite te tauraro o -\frac{145}{144} me \frac{1}{144}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$x^{2}-\frac{1}{6}x+\frac{-145+1}{144}=0$$

Tāpirihia te -145 ki te 1, ka -144.

$$x^{2}-\frac{1}{6}x+\frac{-144}{144}=0$$

Whakawehea te -144 ki te 144, kia riro ko -1.

$$x^{2}-\frac{1}{6}x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -\frac{1}{6} mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\left(-\frac{1}{6}\right)^{2}-4\left(-1\right)}}{2}$$

Pūruatia -\frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\frac{1}{36}-4\left(-1\right)}}{2}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\frac{1}{36}+4}}{2}$$

Tāpiri \frac{1}{36} ki te 4.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\frac{145}{36}}}{2}$$

Tuhia te pūtakerua o te \frac{145}{36}.

$$x=\frac{-\left(-\frac{1}{6}\right)±\frac{\sqrt{145}}{6}}{2}$$

Ko te tauaro o -\frac{1}{6} ko \frac{1}{6}.

$$x=\frac{\frac{1}{6}±\frac{\sqrt{145}}{6}}{2}$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{6}±\frac{\sqrt{145}}{6}}{2} ina he tāpiri te ±. Tāpiri \frac{1}{6} ki te \frac{\sqrt{145}}{6}.

$$x=\frac{\sqrt{145}+1}{2\times 6}$$

Whakawehe \frac{1+\sqrt{145}}{6} ki te 2.

$$x=\frac{\sqrt{145}+1}{12}$$

Nā, me whakaoti te whārite x=\frac{\frac{1}{6}±\frac{\sqrt{145}}{6}}{2} ina he tango te ±. Tango \frac{\sqrt{145}}{6} mai i \frac{1}{6}.

$$x=\frac{1-\sqrt{145}}{2\times 6}$$

Whakawehe \frac{1-\sqrt{145}}{6} ki te 2.

$$x=\frac{1-\sqrt{145}}{12}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{145}+1}{12} x=\frac{1-\sqrt{145}}{12}$$

Ngā Upane mō te Whakaoti i te Pūrua

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o x me 6 ko 6x. Whakareatia \frac{1}{x} ki te \frac{6}{6}. Whakareatia \frac{1}{6} ki te \frac{x}{x}.

$$x=\frac{6}{6x}+\frac{x}{6x}$$

Tā te mea he rite te tauraro o \frac{6}{6x} me \frac{x}{6x}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$x=\frac{6+x}{6x}$$

Tangohia te \frac{6+x}{6x} mai i ngā taha e rua.

$$x-\frac{6+x}{6x}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{6x}{6x}.

$$\frac{x\times 6x}{6x}-\frac{6+x}{6x}=0$$

Tā te mea he rite te tauraro o \frac{x\times 6x}{6x} me \frac{6+x}{6x}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x\times 6x-\left(6+x\right)}{6x}=0$$

Mahia ngā whakarea i roto o x\times 6x-\left(6+x\right).

$$\frac{6x^{2}-6-x}{6x}=0$$

Me whakatauwehe ngā kīanga kāore anō i whakatauwehea i roto o \frac{6x^{2}-6-x}{6x}.

$$\frac{6\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{6x}=0$$

Me whakakore tahi te 6 i te taurunga me te tauraro.

$$\frac{\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

Hei kimi i te tauaro o -\frac{1}{12}\sqrt{145}+\frac{1}{12}, kimihia te tauaro o ia taurangi.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}\right)-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

Ko te tauaro o -\frac{1}{12}\sqrt{145} ko \frac{1}{12}\sqrt{145}.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

Hei kimi i te tauaro o \frac{1}{12}\sqrt{145}+\frac{1}{12}, kimihia te tauaro o ia taurangi.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)=0$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o x+\frac{1}{12}\sqrt{145}-\frac{1}{12} ki ia tau o x-\frac{1}{12}\sqrt{145}-\frac{1}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)\sqrt{145}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Whakareatia te \sqrt{145} ki te \sqrt{145}, ka 145.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Pahekotia te x\left(-\frac{1}{12}\right)\sqrt{145} me \frac{1}{12}\sqrt{145}x, ka 0.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Whakareatia te \frac{1}{12} ki te 145, ka \frac{145}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145}{12}\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te \frac{145}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145\left(-1\right)}{12\times 12}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Mahia ngā whakarea i roto i te hautanga \frac{145\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{-145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Ka taea te hautanga \frac{-145}{144} te tuhi anō ko -\frac{145}{144} mā te tango i te tohu tōraro.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te \frac{1}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Mahia ngā whakarea i roto i te hautanga \frac{1\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{-1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Ka taea te hautanga \frac{-1}{144} te tuhi anō ko -\frac{1}{144} mā te tango i te tohu tōraro.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Pahekotia te x\left(-\frac{1}{12}\right) me -\frac{1}{12}x, ka -\frac{1}{6}x.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te -\frac{1}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{-\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Mahia ngā whakarea i roto i te hautanga \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Pahekotia te -\frac{1}{144}\sqrt{145} me \frac{1}{144}\sqrt{145}, ka 0.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Me whakarea te -\frac{1}{12} ki te -\frac{1}{12} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{-\left(-1\right)}{12\times 12}=0$$

Mahia ngā whakarea i roto i te hautanga \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{1}{144}=0$$

Tā te mea he rite te tauraro o -\frac{145}{144} me \frac{1}{144}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$x^{2}-\frac{1}{6}x+\frac{-145+1}{144}=0$$

Tāpirihia te -145 ki te 1, ka -144.

$$x^{2}-\frac{1}{6}x+\frac{-144}{144}=0$$

Whakawehea te -144 ki te 144, kia riro ko -1.

$$x^{2}-\frac{1}{6}x-1=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}-\frac{1}{6}x=1$$

Whakawehea te -\frac{1}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{12}. Nā, tāpiria te pūrua o te -\frac{1}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=1+\left(-\frac{1}{12}\right)^{2}$$

Pūruatia -\frac{1}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=1+\frac{1}{144}$$

Tāpiri 1 ki te \frac{1}{144}.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{145}{144}$$

Tauwehea x^{2}-\frac{1}{6}x+\frac{1}{144}. 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}.

$$\left(x-\frac{1}{12}\right)^{2}=\frac{145}{144}$$

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

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{\frac{145}{144}}$$

Whakarūnātia.

$$x-\frac{1}{12}=\frac{\sqrt{145}}{12} x-\frac{1}{12}=-\frac{\sqrt{145}}{12}$$

Me tāpiri \frac{1}{12} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{145}+1}{12} x=\frac{1-\sqrt{145}}{12}$$