Whakaoti i te 04+20/10=120/x+120/x+10

$$04 + \frac { 20 } { 10 } = \frac { 120 } { x } + \frac { 120 } { x + 10 }$$

Whakaoti mō x

$$x=5\sqrt{145}+55\approx 115.207972894$$

$$x=55-5\sqrt{145}\approx -5.207972894$$

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 10,x,x+10.

$$0\times 4\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Whakareatia te 0 ki te 4, ka 0.

$$0\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Whakareatia te 0 ki te 10, ka 0.

$$0x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$0+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

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

$$0+\left(x^{2}+10x\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+10x ki te 20.

$$0+20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Ko te tau i tāpiria he kore ka hua koia tonu.

$$20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x+100 ki te 120.

$$20x^{2}+200x=1200x+12000+10x\times 120$$

Whakareatia te 10 ki te 120, ka 1200.

$$20x^{2}+200x=1200x+12000+1200x$$

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

$$20x^{2}+200x=2400x+12000$$

Tangohia te 2400x mai i ngā taha e rua.

$$20x^{2}+200x-2400x=12000$$

Pahekotia te 200x me -2400x, ka -2200x.

$$20x^{2}-2200x=12000$$

Tangohia te 12000 mai i ngā taha e rua.

$$20x^{2}-2200x-12000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 20 mō a, -2200 mō b, me -12000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2200\right)±\sqrt{\left(-2200\right)^{2}-4\times 20\left(-12000\right)}}{2\times 20}$$

Pūrua -2200.

$$x=\frac{-\left(-2200\right)±\sqrt{4840000-4\times 20\left(-12000\right)}}{2\times 20}$$

Whakareatia -4 ki te 20.

$$x=\frac{-\left(-2200\right)±\sqrt{4840000-80\left(-12000\right)}}{2\times 20}$$

Whakareatia -80 ki te -12000.

$$x=\frac{-\left(-2200\right)±\sqrt{4840000+960000}}{2\times 20}$$

Tāpiri 4840000 ki te 960000.

$$x=\frac{-\left(-2200\right)±\sqrt{5800000}}{2\times 20}$$

Tuhia te pūtakerua o te 5800000.

$$x=\frac{-\left(-2200\right)±200\sqrt{145}}{2\times 20}$$

Ko te tauaro o -2200 ko 2200.

$$x=\frac{2200±200\sqrt{145}}{2\times 20}$$

Whakareatia 2 ki te 20.

$$x=\frac{2200±200\sqrt{145}}{40}$$

Nā, me whakaoti te whārite x=\frac{2200±200\sqrt{145}}{40} ina he tāpiri te ±. Tāpiri 2200 ki te 200\sqrt{145}.

$$x=\frac{200\sqrt{145}+2200}{40}$$

Whakawehe 2200+200\sqrt{145} ki te 40.

$$x=5\sqrt{145}+55$$

Nā, me whakaoti te whārite x=\frac{2200±200\sqrt{145}}{40} ina he tango te ±. Tango 200\sqrt{145} mai i 2200.

$$x=\frac{2200-200\sqrt{145}}{40}$$

Whakawehe 2200-200\sqrt{145} ki te 40.

$$x=55-5\sqrt{145}$$

Kua oti te whārite te whakatau.

$$x=5\sqrt{145}+55 x=55-5\sqrt{145}$$

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 10,x,x+10.

$$0\times 4\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Whakareatia te 0 ki te 4, ka 0.

$$0\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Whakareatia te 0 ki te 10, ka 0.

$$0x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$0+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

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

$$0+\left(x^{2}+10x\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+10x ki te 20.

$$0+20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Ko te tau i tāpiria he kore ka hua koia tonu.

$$20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x+100 ki te 120.

$$20x^{2}+200x=1200x+12000+10x\times 120$$

Whakareatia te 10 ki te 120, ka 1200.

$$20x^{2}+200x=1200x+12000+1200x$$

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

$$20x^{2}+200x=2400x+12000$$

Tangohia te 2400x mai i ngā taha e rua.

$$20x^{2}+200x-2400x=12000$$

Pahekotia te 200x me -2400x, ka -2200x.

$$20x^{2}-2200x=12000$$

Whakawehea ngā taha e rua ki te 20.

$$\frac{20x^{2}-2200x}{20}=\frac{12000}{20}$$

Mā te whakawehe ki te 20 ka wetekia te whakareanga ki te 20.

$$x^{2}+\left(-\frac{2200}{20}\right)x=\frac{12000}{20}$$

Whakawehe -2200 ki te 20.

$$x^{2}-110x=\frac{12000}{20}$$

Whakawehe 12000 ki te 20.

$$x^{2}-110x=600$$

Whakawehea te -110, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -55. Nā, tāpiria te pūrua o te -55 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-110x+\left(-55\right)^{2}=600+\left(-55\right)^{2}$$

Pūrua -55.

$$x^{2}-110x+3025=600+3025$$

Tāpiri 600 ki te 3025.

$$x^{2}-110x+3025=3625$$

Tauwehea x^{2}-110x+3025. 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}.

$$\left(x-55\right)^{2}=3625$$

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

$$\sqrt{\left(x-55\right)^{2}}=\sqrt{3625}$$

Whakarūnātia.

$$x-55=5\sqrt{145} x-55=-5\sqrt{145}$$

Me tāpiri 55 ki ngā taha e rua o te whārite.

$$x=5\sqrt{145}+55 x=55-5\sqrt{145}$$