Whakaoti i te 1500/x-1500/x+250=1/2

$$\frac { 1500 } { x } - \frac { 1500 } { x + 250 } = \frac { 1 } { 2 }$$

Whakaoti mō x

$$x=-1000$$

$$x=750$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -250,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x\left(x+250\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+250,2.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+500 ki te 1500.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

Whakareatia te 2 ki te 1500, ka 3000.

$$3000x+750000-3000x=x\left(x+250\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+250.

$$3000x+750000-3000x=x^{2}+250x$$

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

$$3000x+750000-3000x-x^{2}=250x$$

Tangohia te 250x mai i ngā taha e rua.

$$3000x+750000-3000x-x^{2}-250x=0$$

Pahekotia te 3000x me -250x, ka 2750x.

$$2750x+750000-3000x-x^{2}=0$$

Pahekotia te 2750x me -3000x, ka -250x.

$$-250x+750000-x^{2}=0$$

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

$$-x^{2}-250x+750000=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+750000. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-250 ab=-750000=-750000$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -750000.

$$1,-750000 2,-375000 3,-250000 4,-187500 5,-150000 6,-125000 8,-93750 10,-75000 12,-62500 15,-50000 16,-46875 20,-37500 24,-31250 25,-30000 30,-25000 40,-18750 48,-15625 50,-15000 60,-12500 75,-10000 80,-9375 100,-7500 120,-6250 125,-6000 150,-5000 200,-3750 240,-3125 250,-3000 300,-2500 375,-2000 400,-1875 500,-1500 600,-1250 625,-1200 750,-1000$$

Tātaihia te tapeke mō ia takirua.

$$1-750000=-749999 2-375000=-374998 3-250000=-249997 4-187500=-187496 5-150000=-149995 6-125000=-124994 8-93750=-93742 10-75000=-74990 12-62500=-62488 15-50000=-49985 16-46875=-46859 20-37500=-37480 24-31250=-31226 25-30000=-29975 30-25000=-24970 40-18750=-18710 48-15625=-15577 50-15000=-14950 60-12500=-12440 75-10000=-9925 80-9375=-9295 100-7500=-7400 120-6250=-6130 125-6000=-5875 150-5000=-4850 200-3750=-3550 240-3125=-2885 250-3000=-2750 300-2500=-2200 375-2000=-1625 400-1875=-1475 500-1500=-1000 600-1250=-650 625-1200=-575 750-1000=-250$$

Ko te otinga te takirua ka hoatu i te tapeke 250.

$$a=-750 b=1000$$

Tuhia anō te -x^{2}-250x+750000 hei \left(-x^{2}-750x\right)+\left(1000x+750000\right).

$$\left(-x^{2}-750x\right)+\left(1000x+750000\right)$$

Tauwehea te x i te tuatahi me te 1000 i te rōpū tuarua.

$$x\left(x-750\right)+1000\left(x-750\right)$$

Whakatauwehea atu te kīanga pātahi x-750 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-750\right)\left(x+1000\right)$$

Hei kimi otinga whārite, me whakaoti te x-750=0 me te x+1000=0.

$$x=750 x=-1000$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -250,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x\left(x+250\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+250,2.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+500 ki te 1500.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

Whakareatia te 2 ki te 1500, ka 3000.

$$3000x+750000-3000x=x\left(x+250\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+250.

$$3000x+750000-3000x=x^{2}+250x$$

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

$$3000x+750000-3000x-x^{2}=250x$$

Tangohia te 250x mai i ngā taha e rua.

$$3000x+750000-3000x-x^{2}-250x=0$$

Pahekotia te 3000x me -250x, ka 2750x.

$$2750x+750000-3000x-x^{2}=0$$

Pahekotia te 2750x me -3000x, ka -250x.

$$-250x+750000-x^{2}=0$$

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.

$$-x^{2}-250x+750000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -250 mō b, me 750000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-250\right)±\sqrt{\left(-250\right)^{2}-4\left(-1\right)\times 750000}}{2\left(-1\right)}$$

Pūrua -250.

$$x=\frac{-\left(-250\right)±\sqrt{62500-4\left(-1\right)\times 750000}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-250\right)±\sqrt{62500+4\times 750000}}{2\left(-1\right)}$$

Whakareatia 4 ki te 750000.

$$x=\frac{-\left(-250\right)±\sqrt{62500+3000000}}{2\left(-1\right)}$$

Tāpiri 62500 ki te 3000000.

$$x=\frac{-\left(-250\right)±\sqrt{3062500}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 3062500.

$$x=\frac{-\left(-250\right)±1750}{2\left(-1\right)}$$

Ko te tauaro o -250 ko 250.

$$x=\frac{250±1750}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{250±1750}{-2}$$

Nā, me whakaoti te whārite x=\frac{250±1750}{-2} ina he tāpiri te ±. Tāpiri 250 ki te 1750.

$$x=\frac{2000}{-2}$$

Whakawehe 2000 ki te -2.

$$x=-1000$$

Nā, me whakaoti te whārite x=\frac{250±1750}{-2} ina he tango te ±. Tango 1750 mai i 250.

$$x=-\frac{1500}{-2}$$

Whakawehe -1500 ki te -2.

$$x=750$$

Kua oti te whārite te whakatau.

$$x=-1000 x=750$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -250,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x\left(x+250\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+250,2.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+500 ki te 1500.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

Whakareatia te 2 ki te 1500, ka 3000.

$$3000x+750000-3000x=x\left(x+250\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+250.

$$3000x+750000-3000x=x^{2}+250x$$

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

$$3000x+750000-3000x-x^{2}=250x$$

Tangohia te 250x mai i ngā taha e rua.

$$3000x+750000-3000x-x^{2}-250x=0$$

Pahekotia te 3000x me -250x, ka 2750x.

$$2750x+750000-3000x-x^{2}=0$$

Tangohia te 750000 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2750x-3000x-x^{2}=-750000$$

Pahekotia te 2750x me -3000x, ka -250x.

$$-250x-x^{2}=-750000$$

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.

$$-x^{2}-250x=-750000$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-250x}{-1}=-\frac{750000}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{250}{-1}\right)x=-\frac{750000}{-1}$$

Whakawehe -250 ki te -1.

$$x^{2}+250x=-\frac{750000}{-1}$$

Whakawehe -750000 ki te -1.

$$x^{2}+250x=750000$$

Whakawehea te 250, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 125. Nā, tāpiria te pūrua o te 125 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+250x+125^{2}=750000+125^{2}$$

Pūrua 125.

$$x^{2}+250x+15625=750000+15625$$

Tāpiri 750000 ki te 15625.

$$x^{2}+250x+15625=765625$$

Tauwehea x^{2}+250x+15625. 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}.

$$\left(x+125\right)^{2}=765625$$

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

$$\sqrt{\left(x+125\right)^{2}}=\sqrt{765625}$$

Whakarūnātia.

$$x+125=875 x+125=-875$$

Me tango 125 mai i ngā taha e rua o te whārite.

$$x=750 x=-1000$$