Whakaoti i te 2x/x-8+3x/x+5-51/6=5

$$\frac{ 2x }{ x-8 } + \frac{ 3x }{ x+5 } - 5 \frac { 1 } { 6 } =5$$

Whakaoti mō x

$$x = \frac{\sqrt{312361} + 99}{62} \approx 10.611171858$$

$$x=\frac{99-\sqrt{312361}}{62}\approx -7.41762347$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,8 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 6\left(x-8\right)\left(x+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-8,x+5,6.

$$\left(6x+30\right)\times 2x+\left(6x-48\right)\times 3x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x+30 ki te 2.

$$\left(12x+60\right)x+\left(6x-48\right)\times 3x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12x+60 ki te x.

$$12x^{2}+60x+\left(6x-48\right)\times 3x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x-48 ki te 3.

$$12x^{2}+60x+\left(18x-144\right)x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 18x-144 ki te x.

$$12x^{2}+60x+18x^{2}-144x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te 12x^{2} me 18x^{2}, ka 30x^{2}.

$$30x^{2}+60x-144x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te 60x me -144x, ka -84x.

$$30x^{2}-84x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakareatia te 5 ki te 6, ka 30.

$$30x^{2}-84x-\left(x-8\right)\left(x+5\right)\left(30+1\right)=30\left(x-8\right)\left(x+5\right)$$

Tāpirihia te 30 ki te 1, ka 31.

$$30x^{2}-84x-\left(x-8\right)\left(x+5\right)\times 31=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-8 ki te x+5 ka whakakotahi i ngā kupu rite.

$$30x^{2}-84x-\left(x^{2}-3x-40\right)\times 31=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-3x-40 ki te 31.

$$30x^{2}-84x-\left(31x^{2}-93x-1240\right)=30\left(x-8\right)\left(x+5\right)$$

Hei kimi i te tauaro o 31x^{2}-93x-1240, kimihia te tauaro o ia taurangi.

$$30x^{2}-84x-31x^{2}+93x+1240=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te 30x^{2} me -31x^{2}, ka -x^{2}.

$$-x^{2}-84x+93x+1240=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te -84x me 93x, ka 9x.

$$-x^{2}+9x+1240=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 30 ki te x-8.

$$-x^{2}+9x+1240=\left(30x-240\right)\left(x+5\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 30x-240 ki te x+5 ka whakakotahi i ngā kupu rite.

$$-x^{2}+9x+1240=30x^{2}-90x-1200$$

Tangohia te 30x^{2} mai i ngā taha e rua.

$$-x^{2}+9x+1240-30x^{2}=-90x-1200$$

Pahekotia te -x^{2} me -30x^{2}, ka -31x^{2}.

$$-31x^{2}+9x+1240=-90x-1200$$

Me tāpiri te 90x ki ngā taha e rua.

$$-31x^{2}+9x+1240+90x=-1200$$

Pahekotia te 9x me 90x, ka 99x.

$$-31x^{2}+99x+1240=-1200$$

Me tāpiri te 1200 ki ngā taha e rua.

$$-31x^{2}+99x+1240+1200=0$$

Tāpirihia te 1240 ki te 1200, ka 2440.

$$-31x^{2}+99x+2440=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -31 mō a, 99 mō b, me 2440 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-99±\sqrt{99^{2}-4\left(-31\right)\times 2440}}{2\left(-31\right)}$$

Pūrua 99.

$$x=\frac{-99±\sqrt{9801-4\left(-31\right)\times 2440}}{2\left(-31\right)}$$

Whakareatia -4 ki te -31.

$$x=\frac{-99±\sqrt{9801+124\times 2440}}{2\left(-31\right)}$$

Whakareatia 124 ki te 2440.

$$x=\frac{-99±\sqrt{9801+302560}}{2\left(-31\right)}$$

Tāpiri 9801 ki te 302560.

$$x=\frac{-99±\sqrt{312361}}{2\left(-31\right)}$$

Whakareatia 2 ki te -31.

$$x=\frac{-99±\sqrt{312361}}{-62}$$

Nā, me whakaoti te whārite x=\frac{-99±\sqrt{312361}}{-62} ina he tāpiri te ±. Tāpiri -99 ki te \sqrt{312361}.

$$x=\frac{\sqrt{312361}-99}{-62}$$

Whakawehe -99+\sqrt{312361} ki te -62.

$$x=\frac{99-\sqrt{312361}}{62}$$

Nā, me whakaoti te whārite x=\frac{-99±\sqrt{312361}}{-62} ina he tango te ±. Tango \sqrt{312361} mai i -99.

$$x=\frac{-\sqrt{312361}-99}{-62}$$

Whakawehe -99-\sqrt{312361} ki te -62.

$$x=\frac{\sqrt{312361}+99}{62}$$

Kua oti te whārite te whakatau.

$$x=\frac{99-\sqrt{312361}}{62} x=\frac{\sqrt{312361}+99}{62}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,8 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 6\left(x-8\right)\left(x+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-8,x+5,6.

$$\left(6x+30\right)\times 2x+\left(6x-48\right)\times 3x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x+30 ki te 2.

$$\left(12x+60\right)x+\left(6x-48\right)\times 3x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 12x+60 ki te x.

$$12x^{2}+60x+\left(6x-48\right)\times 3x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x-48 ki te 3.

$$12x^{2}+60x+\left(18x-144\right)x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 18x-144 ki te x.

$$12x^{2}+60x+18x^{2}-144x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te 12x^{2} me 18x^{2}, ka 30x^{2}.

$$30x^{2}+60x-144x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te 60x me -144x, ka -84x.

$$30x^{2}-84x-\left(x-8\right)\left(x+5\right)\left(5\times 6+1\right)=30\left(x-8\right)\left(x+5\right)$$

Whakareatia te 5 ki te 6, ka 30.

$$30x^{2}-84x-\left(x-8\right)\left(x+5\right)\left(30+1\right)=30\left(x-8\right)\left(x+5\right)$$

Tāpirihia te 30 ki te 1, ka 31.

$$30x^{2}-84x-\left(x-8\right)\left(x+5\right)\times 31=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-8 ki te x+5 ka whakakotahi i ngā kupu rite.

$$30x^{2}-84x-\left(x^{2}-3x-40\right)\times 31=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-3x-40 ki te 31.

$$30x^{2}-84x-\left(31x^{2}-93x-1240\right)=30\left(x-8\right)\left(x+5\right)$$

Hei kimi i te tauaro o 31x^{2}-93x-1240, kimihia te tauaro o ia taurangi.

$$30x^{2}-84x-31x^{2}+93x+1240=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te 30x^{2} me -31x^{2}, ka -x^{2}.

$$-x^{2}-84x+93x+1240=30\left(x-8\right)\left(x+5\right)$$

Pahekotia te -84x me 93x, ka 9x.

$$-x^{2}+9x+1240=30\left(x-8\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 30 ki te x-8.

$$-x^{2}+9x+1240=\left(30x-240\right)\left(x+5\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 30x-240 ki te x+5 ka whakakotahi i ngā kupu rite.

$$-x^{2}+9x+1240=30x^{2}-90x-1200$$

Tangohia te 30x^{2} mai i ngā taha e rua.

$$-x^{2}+9x+1240-30x^{2}=-90x-1200$$

Pahekotia te -x^{2} me -30x^{2}, ka -31x^{2}.

$$-31x^{2}+9x+1240=-90x-1200$$

Me tāpiri te 90x ki ngā taha e rua.

$$-31x^{2}+9x+1240+90x=-1200$$

Pahekotia te 9x me 90x, ka 99x.

$$-31x^{2}+99x+1240=-1200$$

Tangohia te 1240 mai i ngā taha e rua.

$$-31x^{2}+99x=-1200-1240$$

Tangohia te 1240 i te -1200, ka -2440.

$$-31x^{2}+99x=-2440$$

Whakawehea ngā taha e rua ki te -31.

$$\frac{-31x^{2}+99x}{-31}=-\frac{2440}{-31}$$

Mā te whakawehe ki te -31 ka wetekia te whakareanga ki te -31.

$$x^{2}+\frac{99}{-31}x=-\frac{2440}{-31}$$

Whakawehe 99 ki te -31.

$$x^{2}-\frac{99}{31}x=-\frac{2440}{-31}$$

Whakawehe -2440 ki te -31.

$$x^{2}-\frac{99}{31}x=\frac{2440}{31}$$

Whakawehea te -\frac{99}{31}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{99}{62}. Nā, tāpiria te pūrua o te -\frac{99}{62} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{99}{31}x+\left(-\frac{99}{62}\right)^{2}=\frac{2440}{31}+\left(-\frac{99}{62}\right)^{2}$$

Pūruatia -\frac{99}{62} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{99}{31}x+\frac{9801}{3844}=\frac{2440}{31}+\frac{9801}{3844}$$

Tāpiri \frac{2440}{31} ki te \frac{9801}{3844} 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.

$$x^{2}-\frac{99}{31}x+\frac{9801}{3844}=\frac{312361}{3844}$$

Tauwehea x^{2}-\frac{99}{31}x+\frac{9801}{3844}. 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}.

$$\left(x-\frac{99}{62}\right)^{2}=\frac{312361}{3844}$$

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

$$\sqrt{\left(x-\frac{99}{62}\right)^{2}}=\sqrt{\frac{312361}{3844}}$$

Whakarūnātia.

$$x-\frac{99}{62}=\frac{\sqrt{312361}}{62} x-\frac{99}{62}=-\frac{\sqrt{312361}}{62}$$

Me tāpiri \frac{99}{62} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{312361}+99}{62} x=\frac{99-\sqrt{312361}}{62}$$