Whakaoti i te 400/x+20+400/52/x+20+frac{400/53}{x}=11

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

Whakaoti mō x

$$x = -\frac{80}{11} = -7\frac{3}{11} \approx -7.272727273$$

$$x=60$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -20,0 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+20,x.

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

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

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

Whakareatia te 80 ki te 2, ka 160.

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

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

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

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

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

Whakareatia te 80 ki te 3, ka 240.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+20 ki te 240.

$$560x+240x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x^{2}+220x$$

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

$$800x+4800-11x^{2}=220x$$

Tangohia te 220x mai i ngā taha e rua.

$$800x+4800-11x^{2}-220x=0$$

Pahekotia te 800x me -220x, ka 580x.

$$580x+4800-11x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-11x^{2}+580x+4800=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -11x^{2}+ax+bx+4800. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=580 ab=-11\times 4800=-52800$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -52800.

$$-1,52800 -2,26400 -3,17600 -4,13200 -5,10560 -6,8800 -8,6600 -10,5280 -11,4800 -12,4400 -15,3520 -16,3300 -20,2640 -22,2400 -24,2200 -25,2112 -30,1760 -32,1650 -33,1600 -40,1320 -44,1200 -48,1100 -50,1056 -55,960 -60,880 -64,825 -66,800 -75,704 -80,660 -88,600 -96,550 -100,528 -110,480 -120,440 -132,400 -150,352 -160,330 -165,320 -176,300 -192,275 -200,264 -220,240$$

Tātaihia te tapeke mō ia takirua.

$$-1+52800=52799 -2+26400=26398 -3+17600=17597 -4+13200=13196 -5+10560=10555 -6+8800=8794 -8+6600=6592 -10+5280=5270 -11+4800=4789 -12+4400=4388 -15+3520=3505 -16+3300=3284 -20+2640=2620 -22+2400=2378 -24+2200=2176 -25+2112=2087 -30+1760=1730 -32+1650=1618 -33+1600=1567 -40+1320=1280 -44+1200=1156 -48+1100=1052 -50+1056=1006 -55+960=905 -60+880=820 -64+825=761 -66+800=734 -75+704=629 -80+660=580 -88+600=512 -96+550=454 -100+528=428 -110+480=370 -120+440=320 -132+400=268 -150+352=202 -160+330=170 -165+320=155 -176+300=124 -192+275=83 -200+264=64 -220+240=20$$

Ko te otinga te takirua ka hoatu i te tapeke 580.

$$a=660 b=-80$$

Tuhia anō te -11x^{2}+580x+4800 hei \left(-11x^{2}+660x\right)+\left(-80x+4800\right).

$$\left(-11x^{2}+660x\right)+\left(-80x+4800\right)$$

Tauwehea te 11x i te tuatahi me te 80 i te rōpū tuarua.

$$11x\left(-x+60\right)+80\left(-x+60\right)$$

Whakatauwehea atu te kīanga pātahi -x+60 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+60\right)\left(11x+80\right)$$

Hei kimi otinga whārite, me whakaoti te -x+60=0 me te 11x+80=0.

$$x=60 x=-\frac{80}{11}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -20,0 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+20,x.

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

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

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

Whakareatia te 80 ki te 2, ka 160.

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

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

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

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

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

Whakareatia te 80 ki te 3, ka 240.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+20 ki te 240.

$$560x+240x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x^{2}+220x$$

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

$$800x+4800-11x^{2}=220x$$

Tangohia te 220x mai i ngā taha e rua.

$$800x+4800-11x^{2}-220x=0$$

Pahekotia te 800x me -220x, ka 580x.

$$580x+4800-11x^{2}=0$$

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.

$$-11x^{2}+580x+4800=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -11 mō a, 580 mō b, me 4800 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-580±\sqrt{580^{2}-4\left(-11\right)\times 4800}}{2\left(-11\right)}$$

Pūrua 580.

$$x=\frac{-580±\sqrt{336400-4\left(-11\right)\times 4800}}{2\left(-11\right)}$$

Whakareatia -4 ki te -11.

$$x=\frac{-580±\sqrt{336400+44\times 4800}}{2\left(-11\right)}$$

Whakareatia 44 ki te 4800.

$$x=\frac{-580±\sqrt{336400+211200}}{2\left(-11\right)}$$

Tāpiri 336400 ki te 211200.

$$x=\frac{-580±\sqrt{547600}}{2\left(-11\right)}$$

Tuhia te pūtakerua o te 547600.

$$x=\frac{-580±740}{2\left(-11\right)}$$

Whakareatia 2 ki te -11.

$$x=\frac{-580±740}{-22}$$

Nā, me whakaoti te whārite x=\frac{-580±740}{-22} ina he tāpiri te ±. Tāpiri -580 ki te 740.

$$x=\frac{160}{-22}$$

Whakahekea te hautanga \frac{160}{-22} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-580±740}{-22} ina he tango te ±. Tango 740 mai i -580.

$$x=-\frac{1320}{-22}$$

Whakawehe -1320 ki te -22.

$$x=60$$

Kua oti te whārite te whakatau.

$$x=-\frac{80}{11} x=60$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -20,0 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+20,x.

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

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

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

Whakareatia te 80 ki te 2, ka 160.

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

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

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

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

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

Whakareatia te 80 ki te 3, ka 240.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+20 ki te 240.

$$560x+240x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x\left(x+20\right)$$

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

$$800x+4800=11x^{2}+220x$$

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

$$800x+4800-11x^{2}=220x$$

Tangohia te 220x mai i ngā taha e rua.

$$800x+4800-11x^{2}-220x=0$$

Pahekotia te 800x me -220x, ka 580x.

$$580x+4800-11x^{2}=0$$

Tangohia te 4800 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$580x-11x^{2}=-4800$$

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.

$$-11x^{2}+580x=-4800$$

Whakawehea ngā taha e rua ki te -11.

$$\frac{-11x^{2}+580x}{-11}=-\frac{4800}{-11}$$

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

$$x^{2}+\frac{580}{-11}x=-\frac{4800}{-11}$$

Whakawehe 580 ki te -11.

$$x^{2}-\frac{580}{11}x=-\frac{4800}{-11}$$

Whakawehe -4800 ki te -11.

$$x^{2}-\frac{580}{11}x=\frac{4800}{11}$$

Whakawehea te -\frac{580}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{290}{11}. Nā, tāpiria te pūrua o te -\frac{290}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{580}{11}x+\left(-\frac{290}{11}\right)^{2}=\frac{4800}{11}+\left(-\frac{290}{11}\right)^{2}$$

Pūruatia -\frac{290}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{580}{11}x+\frac{84100}{121}=\frac{4800}{11}+\frac{84100}{121}$$

Tāpiri \frac{4800}{11} ki te \frac{84100}{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.

$$x^{2}-\frac{580}{11}x+\frac{84100}{121}=\frac{136900}{121}$$

Tauwehea x^{2}-\frac{580}{11}x+\frac{84100}{121}. 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}.

$$\left(x-\frac{290}{11}\right)^{2}=\frac{136900}{121}$$

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

$$\sqrt{\left(x-\frac{290}{11}\right)^{2}}=\sqrt{\frac{136900}{121}}$$

Whakarūnātia.

$$x-\frac{290}{11}=\frac{370}{11} x-\frac{290}{11}=-\frac{370}{11}$$

Me tāpiri \frac{290}{11} ki ngā taha e rua o te whārite.

$$x=60 x=-\frac{80}{11}$$