Whakaoti i te 1/6(4x+5)-2/3(2x+7)=3

$$\frac{ 1 }{ 6 } (4x+5) \frac{ -2 }{ 3 } (2x+7)=3$$

Whakaoti mō x (complex solution)

$$x=\frac{-19+3\sqrt{15}i}{8}\approx -2.375+1.452368755i$$

$$x=\frac{-3\sqrt{15}i-19}{8}\approx -2.375-1.452368755i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te hautanga \frac{-2}{3} te tuhi anō ko -\frac{2}{3} mā te tango i te tohu tōraro.

$$\frac{1}{6}\left(4x+5\right)\left(-\frac{2}{3}\right)\left(2x+7\right)=3$$

Whakareatia te \frac{1}{6} ki te -\frac{2}{3}, ka -\frac{1}{9}.

$$-\frac{1}{9}\left(4x+5\right)\left(2x+7\right)=3$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{9} ki te 4x+5.

$$\left(-\frac{4}{9}x-\frac{5}{9}\right)\left(2x+7\right)=3$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -\frac{4}{9}x-\frac{5}{9} ki te 2x+7 ka whakakotahi i ngā kupu rite.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}=3$$

Tangohia te 3 mai i ngā taha e rua.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}-3=0$$

Tangohia te 3 i te -\frac{35}{9}, ka -\frac{62}{9}.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{62}{9}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{8}{9} mō a, -\frac{38}{9} mō b, me -\frac{62}{9} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\left(-\frac{38}{9}\right)^{2}-4\left(-\frac{8}{9}\right)\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Pūruatia -\frac{38}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444}{81}-4\left(-\frac{8}{9}\right)\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Whakareatia -4 ki te -\frac{8}{9}.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444}{81}+\frac{32}{9}\left(-\frac{62}{9}\right)}}{2\left(-\frac{8}{9}\right)}$$

Whakareatia \frac{32}{9} ki te -\frac{62}{9} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{\frac{1444-1984}{81}}}{2\left(-\frac{8}{9}\right)}$$

Tāpiri \frac{1444}{81} ki te -\frac{1984}{81} 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.

$$x=\frac{-\left(-\frac{38}{9}\right)±\sqrt{-\frac{20}{3}}}{2\left(-\frac{8}{9}\right)}$$

Tuhia te pūtakerua o te -\frac{20}{3}.

$$x=\frac{-\left(-\frac{38}{9}\right)±\frac{2\sqrt{15}i}{3}}{2\left(-\frac{8}{9}\right)}$$

Ko te tauaro o -\frac{38}{9} ko \frac{38}{9}.

$$x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{2\left(-\frac{8}{9}\right)}$$

Whakareatia 2 ki te -\frac{8}{9}.

$$x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}}$$

Nā, me whakaoti te whārite x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}} ina he tāpiri te ±. Tāpiri \frac{38}{9} ki te \frac{2i\sqrt{15}}{3}.

$$x=\frac{\frac{2\sqrt{15}i}{3}+\frac{38}{9}}{-\frac{16}{9}}$$

Whakawehe \frac{38}{9}+\frac{2i\sqrt{15}}{3} ki te -\frac{16}{9} mā te whakarea \frac{38}{9}+\frac{2i\sqrt{15}}{3} ki te tau huripoki o -\frac{16}{9}.

$$x=\frac{-3\sqrt{15}i-19}{8}$$

Nā, me whakaoti te whārite x=\frac{\frac{38}{9}±\frac{2\sqrt{15}i}{3}}{-\frac{16}{9}} ina he tango te ±. Tango \frac{2i\sqrt{15}}{3} mai i \frac{38}{9}.

$$x=\frac{-\frac{2\sqrt{15}i}{3}+\frac{38}{9}}{-\frac{16}{9}}$$

Whakawehe \frac{38}{9}-\frac{2i\sqrt{15}}{3} ki te -\frac{16}{9} mā te whakarea \frac{38}{9}-\frac{2i\sqrt{15}}{3} ki te tau huripoki o -\frac{16}{9}.

$$x=\frac{-19+3\sqrt{15}i}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3\sqrt{15}i-19}{8} x=\frac{-19+3\sqrt{15}i}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ka taea te hautanga \frac{-2}{3} te tuhi anō ko -\frac{2}{3} mā te tango i te tohu tōraro.

$$\frac{1}{6}\left(4x+5\right)\left(-\frac{2}{3}\right)\left(2x+7\right)=3$$

Whakareatia te \frac{1}{6} ki te -\frac{2}{3}, ka -\frac{1}{9}.

$$-\frac{1}{9}\left(4x+5\right)\left(2x+7\right)=3$$

Whakamahia te āhuatanga tohatoha hei whakarea te -\frac{1}{9} ki te 4x+5.

$$\left(-\frac{4}{9}x-\frac{5}{9}\right)\left(2x+7\right)=3$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -\frac{4}{9}x-\frac{5}{9} ki te 2x+7 ka whakakotahi i ngā kupu rite.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x-\frac{35}{9}=3$$

Me tāpiri te \frac{35}{9} ki ngā taha e rua.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x=3+\frac{35}{9}$$

Tāpirihia te 3 ki te \frac{35}{9}, ka \frac{62}{9}.

$$-\frac{8}{9}x^{2}-\frac{38}{9}x=\frac{62}{9}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{8}{9}x^{2}-\frac{38}{9}x}{-\frac{8}{9}}=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

Mā te whakawehe ki te -\frac{8}{9} ka wetekia te whakareanga ki te -\frac{8}{9}.

$$x^{2}+\left(-\frac{\frac{38}{9}}{-\frac{8}{9}}\right)x=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

Whakawehe -\frac{38}{9} ki te -\frac{8}{9} mā te whakarea -\frac{38}{9} ki te tau huripoki o -\frac{8}{9}.

$$x^{2}+\frac{19}{4}x=\frac{\frac{62}{9}}{-\frac{8}{9}}$$

Whakawehe \frac{62}{9} ki te -\frac{8}{9} mā te whakarea \frac{62}{9} ki te tau huripoki o -\frac{8}{9}.

$$x^{2}+\frac{19}{4}x=-\frac{31}{4}$$

Whakawehea te \frac{19}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{19}{8}. Nā, tāpiria te pūrua o te \frac{19}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{19}{4}x+\left(\frac{19}{8}\right)^{2}=-\frac{31}{4}+\left(\frac{19}{8}\right)^{2}$$

Pūruatia \frac{19}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=-\frac{31}{4}+\frac{361}{64}$$

Tāpiri -\frac{31}{4} ki te \frac{361}{64} 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.

$$x^{2}+\frac{19}{4}x+\frac{361}{64}=-\frac{135}{64}$$

Tauwehea te x^{2}+\frac{19}{4}x+\frac{361}{64}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{19}{8}\right)^{2}=-\frac{135}{64}$$

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

$$\sqrt{\left(x+\frac{19}{8}\right)^{2}}=\sqrt{-\frac{135}{64}}$$

Whakarūnātia.

$$x+\frac{19}{8}=\frac{3\sqrt{15}i}{8} x+\frac{19}{8}=-\frac{3\sqrt{15}i}{8}$$

Me tango \frac{19}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{-19+3\sqrt{15}i}{8} x=\frac{-3\sqrt{15}i-19}{8}$$