Whakaoti i te 400/x+400div5*2/x+400div5*3/x-20=11

$$\frac{ 400 }{ x } + \frac{ 400 \div 5 \times 2 }{ x } + \frac{ 400 \div 5 \times 3 }{ x-20 } =11$$

Whakaoti mō x

$$x=80$$

$$x = \frac{140}{11} = 12\frac{8}{11} \approx 12.727272727$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,20 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-20\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-20.

$$\left(x-20\right)\times 400+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-20 ki te 400.

$$400x-8000+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakawehea te 400 ki te 5, kia riro ko 80.

$$400x-8000+\left(x-20\right)\times 80\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakareatia te 80 ki te 2, ka 160.

$$400x-8000+\left(x-20\right)\times 160+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-20 ki te 160.

$$400x-8000+160x-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Pahekotia te 400x me 160x, ka 560x.

$$560x-8000-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Tangohia te 3200 i te -8000, ka -11200.

$$560x-11200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakawehea te 400 ki te 5, kia riro ko 80.

$$560x-11200+x\times 80\times 3=11x\left(x-20\right)$$

Whakareatia te 80 ki te 3, ka 240.

$$560x-11200+x\times 240=11x\left(x-20\right)$$

Pahekotia te 560x me x\times 240, ka 800x.

$$800x-11200=11x\left(x-20\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 11x ki te x-20.

$$800x-11200=11x^{2}-220x$$

Tangohia te 11x^{2} mai i ngā taha e rua.

$$800x-11200-11x^{2}=-220x$$

Me tāpiri te 220x ki ngā taha e rua.

$$800x-11200-11x^{2}+220x=0$$

Pahekotia te 800x me 220x, ka 1020x.

$$1020x-11200-11x^{2}=0$$

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.

$$-11x^{2}+1020x-11200=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -11 mō a, 1020 mō b, me -11200 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1020±\sqrt{1020^{2}-4\left(-11\right)\left(-11200\right)}}{2\left(-11\right)}$$

Pūrua 1020.

$$x=\frac{-1020±\sqrt{1040400-4\left(-11\right)\left(-11200\right)}}{2\left(-11\right)}$$

Whakareatia -4 ki te -11.

$$x=\frac{-1020±\sqrt{1040400+44\left(-11200\right)}}{2\left(-11\right)}$$

Whakareatia 44 ki te -11200.

$$x=\frac{-1020±\sqrt{1040400-492800}}{2\left(-11\right)}$$

Tāpiri 1040400 ki te -492800.

$$x=\frac{-1020±\sqrt{547600}}{2\left(-11\right)}$$

Tuhia te pūtakerua o te 547600.

$$x=\frac{-1020±740}{2\left(-11\right)}$$

Whakareatia 2 ki te -11.

$$x=\frac{-1020±740}{-22}$$

Nā, me whakaoti te whārite x=\frac{-1020±740}{-22} ina he tāpiri te ±. Tāpiri -1020 ki te 740.

$$x=-\frac{280}{-22}$$

Whakahekea te hautanga \frac{-280}{-22} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{140}{11}$$

Nā, me whakaoti te whārite x=\frac{-1020±740}{-22} ina he tango te ±. Tango 740 mai i -1020.

$$x=-\frac{1760}{-22}$$

Whakawehe -1760 ki te -22.

$$x=80$$

Kua oti te whārite te whakatau.

$$x=\frac{140}{11} x=80$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,20 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-20\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-20.

$$\left(x-20\right)\times 400+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-20 ki te 400.

$$400x-8000+\left(x-20\right)\times \frac{400}{5}\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakawehea te 400 ki te 5, kia riro ko 80.

$$400x-8000+\left(x-20\right)\times 80\times 2+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakareatia te 80 ki te 2, ka 160.

$$400x-8000+\left(x-20\right)\times 160+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-20 ki te 160.

$$400x-8000+160x-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Pahekotia te 400x me 160x, ka 560x.

$$560x-8000-3200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Tangohia te 3200 i te -8000, ka -11200.

$$560x-11200+x\times \frac{400}{5}\times 3=11x\left(x-20\right)$$

Whakawehea te 400 ki te 5, kia riro ko 80.

$$560x-11200+x\times 80\times 3=11x\left(x-20\right)$$

Whakareatia te 80 ki te 3, ka 240.

$$560x-11200+x\times 240=11x\left(x-20\right)$$

Pahekotia te 560x me x\times 240, ka 800x.

$$800x-11200=11x\left(x-20\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 11x ki te x-20.

$$800x-11200=11x^{2}-220x$$

Tangohia te 11x^{2} mai i ngā taha e rua.

$$800x-11200-11x^{2}=-220x$$

Me tāpiri te 220x ki ngā taha e rua.

$$800x-11200-11x^{2}+220x=0$$

Pahekotia te 800x me 220x, ka 1020x.

$$1020x-11200-11x^{2}=0$$

Me tāpiri te 11200 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$1020x-11x^{2}=11200$$

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.

$$-11x^{2}+1020x=11200$$

Whakawehea ngā taha e rua ki te -11.

$$\frac{-11x^{2}+1020x}{-11}=\frac{11200}{-11}$$

Mā te whakawehe ki te -11 ka wetekia te whakareanga ki te -11.

$$x^{2}+\frac{1020}{-11}x=\frac{11200}{-11}$$

Whakawehe 1020 ki te -11.

$$x^{2}-\frac{1020}{11}x=\frac{11200}{-11}$$

Whakawehe 11200 ki te -11.

$$x^{2}-\frac{1020}{11}x=-\frac{11200}{11}$$

Whakawehea te -\frac{1020}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{510}{11}. Nā, tāpiria te pūrua o te -\frac{510}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1020}{11}x+\left(-\frac{510}{11}\right)^{2}=-\frac{11200}{11}+\left(-\frac{510}{11}\right)^{2}$$

Pūruatia -\frac{510}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1020}{11}x+\frac{260100}{121}=-\frac{11200}{11}+\frac{260100}{121}$$

Tāpiri -\frac{11200}{11} ki te \frac{260100}{121} 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.

$$x^{2}-\frac{1020}{11}x+\frac{260100}{121}=\frac{136900}{121}$$

Tauwehea te x^{2}-\frac{1020}{11}x+\frac{260100}{121}. 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}.

$$\left(x-\frac{510}{11}\right)^{2}=\frac{136900}{121}$$

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

$$\sqrt{\left(x-\frac{510}{11}\right)^{2}}=\sqrt{\frac{136900}{121}}$$

Whakarūnātia.

$$x-\frac{510}{11}=\frac{370}{11} x-\frac{510}{11}=-\frac{370}{11}$$

Me tāpiri \frac{510}{11} ki ngā taha e rua o te whārite.

$$x=80 x=\frac{140}{11}$$