Whakaoti i te 32/3x1/6-3/4left(2x+18right)=-4

$$3 \frac{ 2 }{ 3x } \frac{ 1 }{ 6 } - \frac{ 3 }{ 4 } \left( 2x+18 \right) = -4$$

Whakaoti mō x

$$x=\frac{\sqrt{41}}{2}-\frac{19}{6}\approx 0.034895452$$

$$x=-\frac{\sqrt{41}}{2}-\frac{19}{6}\approx -6.368228785$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12x, arā, te tauraro pātahi he tino iti rawa te kitea o 3x,6,4.

$$3\times 4\times 2\times \frac{1}{6}-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te 3 ki te 4, ka 12.

$$12\times 2\times \frac{1}{6}-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te 12 ki te 2, ka 24.

$$24\times \frac{1}{6}-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te 24 ki te \frac{1}{6}, ka 4.

$$4-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te -\frac{3}{4} ki te 12, ka -9.

$$4-9\left(2x+18\right)x=-48x$$

Whakamahia te āhuatanga tohatoha hei whakarea te -9 ki te 2x+18.

$$4+\left(-18x-162\right)x=-48x$$

Whakamahia te āhuatanga tohatoha hei whakarea te -18x-162 ki te x.

$$4-18x^{2}-162x=-48x$$

Me tāpiri te 48x ki ngā taha e rua.

$$4-18x^{2}-162x+48x=0$$

Pahekotia te -162x me 48x, ka -114x.

$$4-18x^{2}-114x=0$$

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.

$$-18x^{2}-114x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -18 mō a, -114 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-114\right)±\sqrt{\left(-114\right)^{2}-4\left(-18\right)\times 4}}{2\left(-18\right)}$$

Pūrua -114.

$$x=\frac{-\left(-114\right)±\sqrt{12996-4\left(-18\right)\times 4}}{2\left(-18\right)}$$

Whakareatia -4 ki te -18.

$$x=\frac{-\left(-114\right)±\sqrt{12996+72\times 4}}{2\left(-18\right)}$$

Whakareatia 72 ki te 4.

$$x=\frac{-\left(-114\right)±\sqrt{12996+288}}{2\left(-18\right)}$$

Tāpiri 12996 ki te 288.

$$x=\frac{-\left(-114\right)±\sqrt{13284}}{2\left(-18\right)}$$

Tuhia te pūtakerua o te 13284.

$$x=\frac{-\left(-114\right)±18\sqrt{41}}{2\left(-18\right)}$$

Ko te tauaro o -114 ko 114.

$$x=\frac{114±18\sqrt{41}}{2\left(-18\right)}$$

Whakareatia 2 ki te -18.

$$x=\frac{114±18\sqrt{41}}{-36}$$

Nā, me whakaoti te whārite x=\frac{114±18\sqrt{41}}{-36} ina he tāpiri te ±. Tāpiri 114 ki te 18\sqrt{41}.

$$x=\frac{18\sqrt{41}+114}{-36}$$

Whakawehe 114+18\sqrt{41} ki te -36.

$$x=-\frac{\sqrt{41}}{2}-\frac{19}{6}$$

Nā, me whakaoti te whārite x=\frac{114±18\sqrt{41}}{-36} ina he tango te ±. Tango 18\sqrt{41} mai i 114.

$$x=\frac{114-18\sqrt{41}}{-36}$$

Whakawehe 114-18\sqrt{41} ki te -36.

$$x=\frac{\sqrt{41}}{2}-\frac{19}{6}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{41}}{2}-\frac{19}{6} x=\frac{\sqrt{41}}{2}-\frac{19}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12x, arā, te tauraro pātahi he tino iti rawa te kitea o 3x,6,4.

$$3\times 4\times 2\times \frac{1}{6}-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te 3 ki te 4, ka 12.

$$12\times 2\times \frac{1}{6}-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te 12 ki te 2, ka 24.

$$24\times \frac{1}{6}-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te 24 ki te \frac{1}{6}, ka 4.

$$4-\frac{3}{4}\left(2x+18\right)\times 12x=-48x$$

Whakareatia te -\frac{3}{4} ki te 12, ka -9.

$$4-9\left(2x+18\right)x=-48x$$

Whakamahia te āhuatanga tohatoha hei whakarea te -9 ki te 2x+18.

$$4+\left(-18x-162\right)x=-48x$$

Whakamahia te āhuatanga tohatoha hei whakarea te -18x-162 ki te x.

$$4-18x^{2}-162x=-48x$$

Me tāpiri te 48x ki ngā taha e rua.

$$4-18x^{2}-162x+48x=0$$

Pahekotia te -162x me 48x, ka -114x.

$$4-18x^{2}-114x=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-18x^{2}-114x=-4$$

Whakawehea ngā taha e rua ki te -18.

$$\frac{-18x^{2}-114x}{-18}=-\frac{4}{-18}$$

Mā te whakawehe ki te -18 ka wetekia te whakareanga ki te -18.

$$x^{2}+\left(-\frac{114}{-18}\right)x=-\frac{4}{-18}$$

Whakahekea te hautanga \frac{-114}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x^{2}+\frac{19}{3}x=-\frac{4}{-18}$$

Whakahekea te hautanga \frac{-4}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{19}{3}x=\frac{2}{9}$$

Whakawehea te \frac{19}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{19}{6}. Nā, tāpiria te pūrua o te \frac{19}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{19}{3}x+\left(\frac{19}{6}\right)^{2}=\frac{2}{9}+\left(\frac{19}{6}\right)^{2}$$

Pūruatia \frac{19}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{19}{3}x+\frac{361}{36}=\frac{2}{9}+\frac{361}{36}$$

Tāpiri \frac{2}{9} ki te \frac{361}{36} 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.

$$x^{2}+\frac{19}{3}x+\frac{361}{36}=\frac{41}{4}$$

Tauwehea x^{2}+\frac{19}{3}x+\frac{361}{36}. 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}.

$$\left(x+\frac{19}{6}\right)^{2}=\frac{41}{4}$$

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

$$\sqrt{\left(x+\frac{19}{6}\right)^{2}}=\sqrt{\frac{41}{4}}$$

Whakarūnātia.

$$x+\frac{19}{6}=\frac{\sqrt{41}}{2} x+\frac{19}{6}=-\frac{\sqrt{41}}{2}$$

Me tango \frac{19}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{41}}{2}-\frac{19}{6} x=-\frac{\sqrt{41}}{2}-\frac{19}{6}$$