Whakaoti i te 0.2(1-s/500)=0.10-200(1-s/1000)/100s-1000

$$0.2 ( 1 - \frac { s } { 500 } ) = 0.10 - \frac { 200 ( 1 - \frac { s } { 1000 } ) } { 100 s - 1000 }$$

Whakaoti mō s

$$s = \frac{5 \sqrt{3001} + 255}{2} \approx 264.453459248$$

$$s=\frac{255-5\sqrt{3001}}{2}\approx -9.453459248$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe s ki 10 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 500\left(s-10\right), arā, te tauraro pātahi he tino iti rawa te kitea o 500,100s-1000.

$$0.2\left(1-\frac{s}{500}\right)\times 500\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te 0.2 ki te 500, ka 100.

$$100\left(1-\frac{s}{500}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100 ki te 1-\frac{s}{500}.

$$\left(100+100\left(-\frac{s}{500}\right)\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 500 i roto i te 100 me te 500.

$$\left(100+\frac{s}{-5}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100+\frac{s}{-5} ki te s-10.

$$100s-1000+\frac{s}{-5}s-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Tuhia te \frac{s}{-5}s hei hautanga kotahi.

$$100s-1000+\frac{ss}{-5}-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa -5 i roto i te 10 me te -5.

$$100s-1000+\frac{ss}{-5}-2s=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Pahekotia te 100s me -2s, ka 98s.

$$98s-1000+\frac{ss}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te s ki te s, ka s^{2}.

$$98s-1000+\frac{s^{2}}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te 500 ki te 0.1, ka 50.

$$98s-1000+\frac{s^{2}}{-5}=50\left(s-10\right)-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50 ki te s-10.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te -5 ki te 200, ka -1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1000 ki te 1-\frac{s}{1000}.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000-1000\left(-\frac{s}{1000}\right)$$

Whakareatia te -1000 ki te -1, ka 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+1000\times \frac{s}{1000}$$

Tuhia te 1000\times \frac{s}{1000} hei hautanga kotahi.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+\frac{1000s}{1000}$$

Me whakakore te 1000 me te 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+s$$

Tangohia te 1000 i te -500, ka -1500.

$$98s-1000+\frac{s^{2}}{-5}=50s-1500+s$$

Pahekotia te 50s me s, ka 51s.

$$98s-1000+\frac{s^{2}}{-5}=51s-1500$$

Tangohia te 51s mai i ngā taha e rua.

$$98s-1000+\frac{s^{2}}{-5}-51s=-1500$$

Pahekotia te 98s me -51s, ka 47s.

$$47s-1000+\frac{s^{2}}{-5}=-1500$$

Me tāpiri te 1500 ki ngā taha e rua.

$$47s-1000+\frac{s^{2}}{-5}+1500=0$$

Tāpirihia te -1000 ki te 1500, ka 500.

$$47s+500+\frac{s^{2}}{-5}=0$$

Whakareatia ngā taha e rua o te whārite ki te -5.

$$-235s-2500+s^{2}=0$$

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.

$$s^{2}-235s-2500=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -235 mō b, me -2500 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-\left(-235\right)±\sqrt{\left(-235\right)^{2}-4\left(-2500\right)}}{2}$$

Pūrua -235.

$$s=\frac{-\left(-235\right)±\sqrt{55225-4\left(-2500\right)}}{2}$$

Whakareatia -4 ki te -2500.

$$s=\frac{-\left(-235\right)±\sqrt{55225+10000}}{2}$$

Tāpiri 55225 ki te 10000.

$$s=\frac{-\left(-235\right)±\sqrt{65225}}{2}$$

Tuhia te pūtakerua o te 65225.

$$s=\frac{-\left(-235\right)±5\sqrt{2609}}{2}$$

Ko te tauaro o -235 ko 235.

$$s=\frac{235±5\sqrt{2609}}{2}$$

Nā, me whakaoti te whārite s=\frac{235±5\sqrt{2609}}{2} ina he tāpiri te ±. Tāpiri 235 ki te 5\sqrt{2609}.

$$s=\frac{5\sqrt{2609}+235}{2}$$

Nā, me whakaoti te whārite s=\frac{235±5\sqrt{2609}}{2} ina he tango te ±. Tango 5\sqrt{2609} mai i 235.

$$s=\frac{235-5\sqrt{2609}}{2}$$

Kua oti te whārite te whakatau.

$$s=\frac{5\sqrt{2609}+235}{2} s=\frac{235-5\sqrt{2609}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe s ki 10 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 500\left(s-10\right), arā, te tauraro pātahi he tino iti rawa te kitea o 500,100s-1000.

$$0.2\left(1-\frac{s}{500}\right)\times 500\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te 0.2 ki te 500, ka 100.

$$100\left(1-\frac{s}{500}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100 ki te 1-\frac{s}{500}.

$$\left(100+100\left(-\frac{s}{500}\right)\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 500 i roto i te 100 me te 500.

$$\left(100+\frac{s}{-5}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 100+\frac{s}{-5} ki te s-10.

$$100s-1000+\frac{s}{-5}s-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Tuhia te \frac{s}{-5}s hei hautanga kotahi.

$$100s-1000+\frac{ss}{-5}-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa -5 i roto i te 10 me te -5.

$$100s-1000+\frac{ss}{-5}-2s=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Pahekotia te 100s me -2s, ka 98s.

$$98s-1000+\frac{ss}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te s ki te s, ka s^{2}.

$$98s-1000+\frac{s^{2}}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te 500 ki te 0.1, ka 50.

$$98s-1000+\frac{s^{2}}{-5}=50\left(s-10\right)-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50 ki te s-10.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-5\times 200\left(1-\frac{s}{1000}\right)$$

Whakareatia te -5 ki te 200, ka -1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000\left(1-\frac{s}{1000}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1000 ki te 1-\frac{s}{1000}.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000-1000\left(-\frac{s}{1000}\right)$$

Whakareatia te -1000 ki te -1, ka 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+1000\times \frac{s}{1000}$$

Tuhia te 1000\times \frac{s}{1000} hei hautanga kotahi.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+\frac{1000s}{1000}$$

Me whakakore te 1000 me te 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+s$$

Tangohia te 1000 i te -500, ka -1500.

$$98s-1000+\frac{s^{2}}{-5}=50s-1500+s$$

Pahekotia te 50s me s, ka 51s.

$$98s-1000+\frac{s^{2}}{-5}=51s-1500$$

Tangohia te 51s mai i ngā taha e rua.

$$98s-1000+\frac{s^{2}}{-5}-51s=-1500$$

Pahekotia te 98s me -51s, ka 47s.

$$47s-1000+\frac{s^{2}}{-5}=-1500$$

Me tāpiri te 1000 ki ngā taha e rua.

$$47s+\frac{s^{2}}{-5}=-1500+1000$$

Tāpirihia te -1500 ki te 1000, ka -500.

$$47s+\frac{s^{2}}{-5}=-500$$

Whakareatia ngā taha e rua o te whārite ki te -5.

$$-235s+s^{2}=2500$$

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.

$$s^{2}-235s=2500$$

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

$$s^{2}-235s+\left(-\frac{235}{2}\right)^{2}=2500+\left(-\frac{235}{2}\right)^{2}$$

Pūruatia -\frac{235}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$s^{2}-235s+\frac{55225}{4}=2500+\frac{55225}{4}$$

Tāpiri 2500 ki te \frac{55225}{4}.

$$s^{2}-235s+\frac{55225}{4}=\frac{65225}{4}$$

Tauwehea s^{2}-235s+\frac{55225}{4}. 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}.

$$\left(s-\frac{235}{2}\right)^{2}=\frac{65225}{4}$$

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

$$\sqrt{\left(s-\frac{235}{2}\right)^{2}}=\sqrt{\frac{65225}{4}}$$

Whakarūnātia.

$$s-\frac{235}{2}=\frac{5\sqrt{2609}}{2} s-\frac{235}{2}=-\frac{5\sqrt{2609}}{2}$$

Me tāpiri \frac{235}{2} ki ngā taha e rua o te whārite.

$$s=\frac{5\sqrt{2609}+235}{2} s=\frac{235-5\sqrt{2609}}{2}$$