Whakaoti i te 3x+10/x-2[9x-4/3-3(x/2+7x-6/4)]=frac{7x+5}{2}

$$\frac { 3 x + 10 } { x } - 2 [ \frac { 9 x - 4 } { 3 } - 3 ( \frac { x } { 2 } + \frac { 7 x - 6 } { 4 } ) ] = \frac { 7 x + 5 } { 2 }$$

Whakaoti mō x (complex solution)

$$x=\frac{35+\sqrt{4535}i}{48}\approx 0.729166667+1.402966846i$$

$$x=\frac{-\sqrt{4535}i+35}{48}\approx 0.729166667-1.402966846i$$

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 x,3,2,4.

$$12\left(3x+10\right)-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 3x+10.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 2 me 4 ko 4. Whakareatia \frac{x}{2} ki te \frac{2}{2}.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{2x}{4}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Tā te mea he rite te tauraro o \frac{2x}{4} me \frac{7x-6}{4}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{2x+7x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Whakakotahitia ngā kupu rite i 2x+7x-6.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{9x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Tuhia te 3\times \frac{9x-6}{4} hei hautanga kotahi.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{3\left(9x-6\right)}{4}\right)\times 12x=6x\left(7x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 9x-6.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{27x-18}{4}\right)\times 12x=6x\left(7x+5\right)$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 3 me 4 ko 12. Whakareatia \frac{9x-4}{3} ki te \frac{4}{4}. Whakareatia \frac{27x-18}{4} ki te \frac{3}{3}.

$$36x+120-2\left(\frac{4\left(9x-4\right)}{12}-\frac{3\left(27x-18\right)}{12}\right)\times 12x=6x\left(7x+5\right)$$

Tā te mea he rite te tauraro o \frac{4\left(9x-4\right)}{12} me \frac{3\left(27x-18\right)}{12}, me tango rāua mā te tango i ō raua taurunga.

$$36x+120-2\times \frac{4\left(9x-4\right)-3\left(27x-18\right)}{12}\times 12x=6x\left(7x+5\right)$$

Mahia ngā whakarea i roto o 4\left(9x-4\right)-3\left(27x-18\right).

$$36x+120-2\times \frac{36x-16-81x+54}{12}\times 12x=6x\left(7x+5\right)$$

Whakakotahitia ngā kupu rite i 36x-16-81x+54.

$$36x+120-2\times \frac{-45x+38}{12}\times 12x=6x\left(7x+5\right)$$

Whakareatia te 2 ki te 12, ka 24.

$$36x+120-24\times \frac{-45x+38}{12}x=6x\left(7x+5\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 12 i roto i te 24 me te 12.

$$36x+120-2\left(-45x+38\right)x=6x\left(7x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x ki te 7x+5.

$$36x+120-2\left(-45x+38\right)x=42x^{2}+30x$$

Tangohia te 42x^{2} mai i ngā taha e rua.

$$36x+120-2\left(-45x+38\right)x-42x^{2}=30x$$

Tangohia te 30x mai i ngā taha e rua.

$$36x+120-2\left(-45x+38\right)x-42x^{2}-30x=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te -45x+38.

$$36x+120+\left(90x-76\right)x-42x^{2}-30x=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 90x-76 ki te x.

$$36x+120+90x^{2}-76x-42x^{2}-30x=0$$

Pahekotia te 36x me -76x, ka -40x.

$$-40x+120+90x^{2}-42x^{2}-30x=0$$

Pahekotia te 90x^{2} me -42x^{2}, ka 48x^{2}.

$$-40x+120+48x^{2}-30x=0$$

Pahekotia te -40x me -30x, ka -70x.

$$-70x+120+48x^{2}=0$$

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.

$$48x^{2}-70x+120=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 48 mō a, -70 mō b, me 120 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 48\times 120}}{2\times 48}$$

Pūrua -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 48\times 120}}{2\times 48}$$

Whakareatia -4 ki te 48.

$$x=\frac{-\left(-70\right)±\sqrt{4900-192\times 120}}{2\times 48}$$

Whakareatia -192 ki te 120.

$$x=\frac{-\left(-70\right)±\sqrt{4900-23040}}{2\times 48}$$

Tāpiri 4900 ki te -23040.

$$x=\frac{-\left(-70\right)±\sqrt{-18140}}{2\times 48}$$

Tuhia te pūtakerua o te -18140.

$$x=\frac{-\left(-70\right)±2\sqrt{4535}i}{2\times 48}$$

Ko te tauaro o -70 ko 70.

$$x=\frac{70±2\sqrt{4535}i}{2\times 48}$$

Whakareatia 2 ki te 48.

$$x=\frac{70±2\sqrt{4535}i}{96}$$

Nā, me whakaoti te whārite x=\frac{70±2\sqrt{4535}i}{96} ina he tāpiri te ±. Tāpiri 70 ki te 2i\sqrt{4535}.

$$x=\frac{70+2\sqrt{4535}i}{96}$$

Whakawehe 70+2i\sqrt{4535} ki te 96.

$$x=\frac{35+\sqrt{4535}i}{48}$$

Nā, me whakaoti te whārite x=\frac{70±2\sqrt{4535}i}{96} ina he tango te ±. Tango 2i\sqrt{4535} mai i 70.

$$x=\frac{-2\sqrt{4535}i+70}{96}$$

Whakawehe 70-2i\sqrt{4535} ki te 96.

$$x=\frac{-\sqrt{4535}i+35}{48}$$

Kua oti te whārite te whakatau.

$$x=\frac{35+\sqrt{4535}i}{48} x=\frac{-\sqrt{4535}i+35}{48}$$

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 x,3,2,4.

$$12\left(3x+10\right)-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 3x+10.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 2 me 4 ko 4. Whakareatia \frac{x}{2} ki te \frac{2}{2}.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{2x}{4}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Tā te mea he rite te tauraro o \frac{2x}{4} me \frac{7x-6}{4}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{2x+7x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Whakakotahitia ngā kupu rite i 2x+7x-6.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{9x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Tuhia te 3\times \frac{9x-6}{4} hei hautanga kotahi.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{3\left(9x-6\right)}{4}\right)\times 12x=6x\left(7x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 9x-6.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{27x-18}{4}\right)\times 12x=6x\left(7x+5\right)$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 3 me 4 ko 12. Whakareatia \frac{9x-4}{3} ki te \frac{4}{4}. Whakareatia \frac{27x-18}{4} ki te \frac{3}{3}.

$$36x+120-2\left(\frac{4\left(9x-4\right)}{12}-\frac{3\left(27x-18\right)}{12}\right)\times 12x=6x\left(7x+5\right)$$

Tā te mea he rite te tauraro o \frac{4\left(9x-4\right)}{12} me \frac{3\left(27x-18\right)}{12}, me tango rāua mā te tango i ō raua taurunga.

$$36x+120-2\times \frac{4\left(9x-4\right)-3\left(27x-18\right)}{12}\times 12x=6x\left(7x+5\right)$$

Mahia ngā whakarea i roto o 4\left(9x-4\right)-3\left(27x-18\right).

$$36x+120-2\times \frac{36x-16-81x+54}{12}\times 12x=6x\left(7x+5\right)$$

Whakakotahitia ngā kupu rite i 36x-16-81x+54.

$$36x+120-2\times \frac{-45x+38}{12}\times 12x=6x\left(7x+5\right)$$

Whakareatia te 2 ki te 12, ka 24.

$$36x+120-24\times \frac{-45x+38}{12}x=6x\left(7x+5\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 12 i roto i te 24 me te 12.

$$36x+120-2\left(-45x+38\right)x=6x\left(7x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x ki te 7x+5.

$$36x+120-2\left(-45x+38\right)x=42x^{2}+30x$$

Tangohia te 42x^{2} mai i ngā taha e rua.

$$36x+120-2\left(-45x+38\right)x-42x^{2}=30x$$

Tangohia te 30x mai i ngā taha e rua.

$$36x+120-2\left(-45x+38\right)x-42x^{2}-30x=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te -45x+38.

$$36x+120+\left(90x-76\right)x-42x^{2}-30x=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 90x-76 ki te x.

$$36x+120+90x^{2}-76x-42x^{2}-30x=0$$

Pahekotia te 36x me -76x, ka -40x.

$$-40x+120+90x^{2}-42x^{2}-30x=0$$

Pahekotia te 90x^{2} me -42x^{2}, ka 48x^{2}.

$$-40x+120+48x^{2}-30x=0$$

Pahekotia te -40x me -30x, ka -70x.

$$-70x+120+48x^{2}=0$$

Tangohia te 120 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-70x+48x^{2}=-120$$

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.

$$48x^{2}-70x=-120$$

Whakawehea ngā taha e rua ki te 48.

$$\frac{48x^{2}-70x}{48}=-\frac{120}{48}$$

Mā te whakawehe ki te 48 ka wetekia te whakareanga ki te 48.

$$x^{2}+\left(-\frac{70}{48}\right)x=-\frac{120}{48}$$

Whakahekea te hautanga \frac{-70}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{35}{24}x=-\frac{120}{48}$$

Whakahekea te hautanga \frac{-120}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 24.

$$x^{2}-\frac{35}{24}x=-\frac{5}{2}$$

Whakawehea te -\frac{35}{24}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{35}{48}. Nā, tāpiria te pūrua o te -\frac{35}{48} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{35}{24}x+\left(-\frac{35}{48}\right)^{2}=-\frac{5}{2}+\left(-\frac{35}{48}\right)^{2}$$

Pūruatia -\frac{35}{48} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=-\frac{5}{2}+\frac{1225}{2304}$$

Tāpiri -\frac{5}{2} ki te \frac{1225}{2304} 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.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=-\frac{4535}{2304}$$

Tauwehea x^{2}-\frac{35}{24}x+\frac{1225}{2304}. 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}.

$$\left(x-\frac{35}{48}\right)^{2}=-\frac{4535}{2304}$$

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

$$\sqrt{\left(x-\frac{35}{48}\right)^{2}}=\sqrt{-\frac{4535}{2304}}$$

Whakarūnātia.

$$x-\frac{35}{48}=\frac{\sqrt{4535}i}{48} x-\frac{35}{48}=-\frac{\sqrt{4535}i}{48}$$

Me tāpiri \frac{35}{48} ki ngā taha e rua o te whārite.

$$x=\frac{35+\sqrt{4535}i}{48} x=\frac{-\sqrt{4535}i+35}{48}$$