Whakaoti i te (1/2-x)*x=[2/7(1-1/5)]:[(1-3/5):(1+frac{2}{5})]

$$( \frac { 1 } { 2 } - x ) \cdot x = [ \frac { 2 } { 7 } ( 1 - \frac { 1 } { 5 } ) ] : [ ( 1 - \frac { 3 } { 5 } ) : ( 1 + \frac { 2 } { 5 } ) ]$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{295}i}{20}+\frac{1}{4}\approx 0.25+0.858778202i$$

$$x=-\frac{\sqrt{295}i}{20}+\frac{1}{4}\approx 0.25-0.858778202i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2}-x ki te x.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(1-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(\frac{5}{5}-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{1}{5}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{5-1}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Tangohia te 1 i te 5, ka 4.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{4}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Me whakarea te \frac{2}{7} ki te \frac{4}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2\times 4}{7\times 5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Mahia ngā whakarea i roto i te hautanga \frac{2\times 4}{7\times 5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5}{5}-\frac{3}{5}}{1+\frac{2}{5}}}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{3}{5}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5-3}{5}}{1+\frac{2}{5}}}$$

Tangohia te 3 i te 5, ka 2.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{1+\frac{2}{5}}}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5}{5}+\frac{2}{5}}}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{2}{5}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5+2}{5}}}$$

Tāpirihia te 5 ki te 2, ka 7.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{7}{5}}}$$

Whakawehe \frac{2}{5} ki te \frac{7}{5} mā te whakarea \frac{2}{5} ki te tau huripoki o \frac{7}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{5}\times \frac{5}{7}}$$

Me whakarea te \frac{2}{5} ki te \frac{5}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2\times 5}{5\times 7}}$$

Me whakakore tahi te 5 i te taurunga me te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{7}}$$

Whakawehe \frac{8}{35} ki te \frac{2}{7} mā te whakarea \frac{8}{35} ki te tau huripoki o \frac{2}{7}.

$$\frac{1}{2}x-x^{2}=\frac{8}{35}\times \frac{7}{2}$$

Me whakarea te \frac{8}{35} ki te \frac{7}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{8\times 7}{35\times 2}$$

Mahia ngā whakarea i roto i te hautanga \frac{8\times 7}{35\times 2}.

$$\frac{1}{2}x-x^{2}=\frac{56}{70}$$

Whakahekea te hautanga \frac{56}{70} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

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

Tangohia te \frac{4}{5} mai i ngā taha e rua.

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

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.

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

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

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

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

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

Whakareatia -4 ki te -1.

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

Whakareatia 4 ki te -\frac{4}{5}.

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

Tāpiri \frac{1}{4} ki te -\frac{16}{5} 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.

$$x=\frac{-\frac{1}{2}±\sqrt{-\frac{59}{20}}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -\frac{59}{20}.

$$x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{-2} ina he tāpiri te ±. Tāpiri -\frac{1}{2} ki te \frac{i\sqrt{295}}{10}.

$$x=\frac{\frac{\sqrt{295}i}{10}-\frac{1}{2}}{-2}$$

Whakawehe -\frac{1}{2}+\frac{i\sqrt{295}}{10} ki te -2.

$$x=-\frac{\sqrt{295}i}{20}+\frac{1}{4}$$

Nā, me whakaoti te whārite x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{-2} ina he tango te ±. Tango \frac{i\sqrt{295}}{10} mai i -\frac{1}{2}.

$$x=\frac{-\frac{\sqrt{295}i}{10}-\frac{1}{2}}{-2}$$

Whakawehe -\frac{1}{2}-\frac{i\sqrt{295}}{10} ki te -2.

$$x=\frac{\sqrt{295}i}{20}+\frac{1}{4}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{295}i}{20}+\frac{1}{4} x=\frac{\sqrt{295}i}{20}+\frac{1}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{2}-x ki te x.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(1-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(\frac{5}{5}-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{1}{5}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{5-1}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Tangohia te 1 i te 5, ka 4.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{4}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Me whakarea te \frac{2}{7} ki te \frac{4}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2\times 4}{7\times 5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Mahia ngā whakarea i roto i te hautanga \frac{2\times 4}{7\times 5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5}{5}-\frac{3}{5}}{1+\frac{2}{5}}}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{3}{5}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5-3}{5}}{1+\frac{2}{5}}}$$

Tangohia te 3 i te 5, ka 2.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{1+\frac{2}{5}}}$$

Me tahuri te 1 ki te hautau \frac{5}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5}{5}+\frac{2}{5}}}$$

Tā te mea he rite te tauraro o \frac{5}{5} me \frac{2}{5}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5+2}{5}}}$$

Tāpirihia te 5 ki te 2, ka 7.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{7}{5}}}$$

Whakawehe \frac{2}{5} ki te \frac{7}{5} mā te whakarea \frac{2}{5} ki te tau huripoki o \frac{7}{5}.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{5}\times \frac{5}{7}}$$

Me whakarea te \frac{2}{5} ki te \frac{5}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2\times 5}{5\times 7}}$$

Me whakakore tahi te 5 i te taurunga me te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{7}}$$

Whakawehe \frac{8}{35} ki te \frac{2}{7} mā te whakarea \frac{8}{35} ki te tau huripoki o \frac{2}{7}.

$$\frac{1}{2}x-x^{2}=\frac{8}{35}\times \frac{7}{2}$$

Me whakarea te \frac{8}{35} ki te \frac{7}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{1}{2}x-x^{2}=\frac{8\times 7}{35\times 2}$$

Mahia ngā whakarea i roto i te hautanga \frac{8\times 7}{35\times 2}.

$$\frac{1}{2}x-x^{2}=\frac{56}{70}$$

Whakahekea te hautanga \frac{56}{70} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

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

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.

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

Whakawehea ngā taha e rua ki te -1.

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

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

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

Whakawehe \frac{1}{2} ki te -1.

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

Whakawehe \frac{4}{5} ki te -1.

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

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{4}{5}+\left(-\frac{1}{4}\right)^{2}$$

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{4}{5}+\frac{1}{16}$$

Tāpiri -\frac{4}{5} ki te \frac{1}{16} 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.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{59}{80}$$

Tauwehea x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{59}{80}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{59}{80}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{\sqrt{295}i}{20} x-\frac{1}{4}=-\frac{\sqrt{295}i}{20}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{295}i}{20}+\frac{1}{4} x=-\frac{\sqrt{295}i}{20}+\frac{1}{4}$$