Whakaoti i te 94+240/x=120/10+120/x+10

$$94 + \frac { 240 } { x } = \frac { 120 } { 10 } + \frac { 120 } { x + 10 }$$

Whakaoti mō x

$$x=\frac{5\sqrt{241}-235}{41}\approx -3.838515281$$

$$x=\frac{-5\sqrt{241}-235}{41}\approx -7.624899353$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 10x\left(x+10\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,10,x+10.

$$10x\left(x+10\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

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

$$\left(10x^{2}+100x\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x^{2}+100x ki te 94.

$$940x^{2}+9400x+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x+100 ki te 240.

$$940x^{2}+9400x+2400x+24000=x\left(x+10\right)\times 120+10x\times 120$$

Pahekotia te 9400x me 2400x, ka 11800x.

$$940x^{2}+11800x+24000=x\left(x+10\right)\times 120+10x\times 120$$

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

$$940x^{2}+11800x+24000=\left(x^{2}+10x\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+10x ki te 120.

$$940x^{2}+11800x+24000=120x^{2}+1200x+10x\times 120$$

Whakareatia te 10 ki te 120, ka 1200.

$$940x^{2}+11800x+24000=120x^{2}+1200x+1200x$$

Pahekotia te 1200x me 1200x, ka 2400x.

$$940x^{2}+11800x+24000=120x^{2}+2400x$$

Tangohia te 120x^{2} mai i ngā taha e rua.

$$940x^{2}+11800x+24000-120x^{2}=2400x$$

Pahekotia te 940x^{2} me -120x^{2}, ka 820x^{2}.

$$820x^{2}+11800x+24000=2400x$$

Tangohia te 2400x mai i ngā taha e rua.

$$820x^{2}+11800x+24000-2400x=0$$

Pahekotia te 11800x me -2400x, ka 9400x.

$$820x^{2}+9400x+24000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 820 mō a, 9400 mō b, me 24000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9400±\sqrt{9400^{2}-4\times 820\times 24000}}{2\times 820}$$

Pūrua 9400.

$$x=\frac{-9400±\sqrt{88360000-4\times 820\times 24000}}{2\times 820}$$

Whakareatia -4 ki te 820.

$$x=\frac{-9400±\sqrt{88360000-3280\times 24000}}{2\times 820}$$

Whakareatia -3280 ki te 24000.

$$x=\frac{-9400±\sqrt{88360000-78720000}}{2\times 820}$$

Tāpiri 88360000 ki te -78720000.

$$x=\frac{-9400±\sqrt{9640000}}{2\times 820}$$

Tuhia te pūtakerua o te 9640000.

$$x=\frac{-9400±200\sqrt{241}}{2\times 820}$$

Whakareatia 2 ki te 820.

$$x=\frac{-9400±200\sqrt{241}}{1640}$$

Nā, me whakaoti te whārite x=\frac{-9400±200\sqrt{241}}{1640} ina he tāpiri te ±. Tāpiri -9400 ki te 200\sqrt{241}.

$$x=\frac{200\sqrt{241}-9400}{1640}$$

Whakawehe -9400+200\sqrt{241} ki te 1640.

$$x=\frac{5\sqrt{241}-235}{41}$$

Nā, me whakaoti te whārite x=\frac{-9400±200\sqrt{241}}{1640} ina he tango te ±. Tango 200\sqrt{241} mai i -9400.

$$x=\frac{-200\sqrt{241}-9400}{1640}$$

Whakawehe -9400-200\sqrt{241} ki te 1640.

$$x=\frac{-5\sqrt{241}-235}{41}$$

Kua oti te whārite te whakatau.

$$x=\frac{5\sqrt{241}-235}{41} x=\frac{-5\sqrt{241}-235}{41}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 10x\left(x+10\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,10,x+10.

$$10x\left(x+10\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

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

$$\left(10x^{2}+100x\right)\times 94+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x^{2}+100x ki te 94.

$$940x^{2}+9400x+\left(10x+100\right)\times 240=x\left(x+10\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x+100 ki te 240.

$$940x^{2}+9400x+2400x+24000=x\left(x+10\right)\times 120+10x\times 120$$

Pahekotia te 9400x me 2400x, ka 11800x.

$$940x^{2}+11800x+24000=x\left(x+10\right)\times 120+10x\times 120$$

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

$$940x^{2}+11800x+24000=\left(x^{2}+10x\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+10x ki te 120.

$$940x^{2}+11800x+24000=120x^{2}+1200x+10x\times 120$$

Whakareatia te 10 ki te 120, ka 1200.

$$940x^{2}+11800x+24000=120x^{2}+1200x+1200x$$

Pahekotia te 1200x me 1200x, ka 2400x.

$$940x^{2}+11800x+24000=120x^{2}+2400x$$

Tangohia te 120x^{2} mai i ngā taha e rua.

$$940x^{2}+11800x+24000-120x^{2}=2400x$$

Pahekotia te 940x^{2} me -120x^{2}, ka 820x^{2}.

$$820x^{2}+11800x+24000=2400x$$

Tangohia te 2400x mai i ngā taha e rua.

$$820x^{2}+11800x+24000-2400x=0$$

Pahekotia te 11800x me -2400x, ka 9400x.

$$820x^{2}+9400x+24000=0$$

Tangohia te 24000 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$820x^{2}+9400x=-24000$$

Whakawehea ngā taha e rua ki te 820.

$$\frac{820x^{2}+9400x}{820}=-\frac{24000}{820}$$

Mā te whakawehe ki te 820 ka wetekia te whakareanga ki te 820.

$$x^{2}+\frac{9400}{820}x=-\frac{24000}{820}$$

Whakahekea te hautanga \frac{9400}{820} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x^{2}+\frac{470}{41}x=-\frac{24000}{820}$$

Whakahekea te hautanga \frac{-24000}{820} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x^{2}+\frac{470}{41}x=-\frac{1200}{41}$$

Whakawehea te \frac{470}{41}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{235}{41}. Nā, tāpiria te pūrua o te \frac{235}{41} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{470}{41}x+\left(\frac{235}{41}\right)^{2}=-\frac{1200}{41}+\left(\frac{235}{41}\right)^{2}$$

Pūruatia \frac{235}{41} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{470}{41}x+\frac{55225}{1681}=-\frac{1200}{41}+\frac{55225}{1681}$$

Tāpiri -\frac{1200}{41} ki te \frac{55225}{1681} 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.

$$x^{2}+\frac{470}{41}x+\frac{55225}{1681}=\frac{6025}{1681}$$

Tauwehea x^{2}+\frac{470}{41}x+\frac{55225}{1681}. 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}.

$$\left(x+\frac{235}{41}\right)^{2}=\frac{6025}{1681}$$

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

$$\sqrt{\left(x+\frac{235}{41}\right)^{2}}=\sqrt{\frac{6025}{1681}}$$

Whakarūnātia.

$$x+\frac{235}{41}=\frac{5\sqrt{241}}{41} x+\frac{235}{41}=-\frac{5\sqrt{241}}{41}$$

Me tango \frac{235}{41} mai i ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{241}-235}{41} x=\frac{-5\sqrt{241}-235}{41}$$