Whakaoti i te 12000=s^2+20s

$$12000 = s ^ { 2 } + 20 s$$

Whakaoti mō s

$$s=-120$$

$$s=100$$

Ngā Upane Whakamahi Tauwehe

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$s^{2}+20s=12000$$

Tangohia te 12000 mai i ngā taha e rua.

$$s^{2}+20s-12000=0$$

Hei whakaoti i te whārite, whakatauwehea te s^{2}+20s-12000 mā te whakamahi i te tātai s^{2}+\left(a+b\right)s+ab=\left(s+a\right)\left(s+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=20 ab=-12000$$

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 -12000.

$$-1,12000 -2,6000 -3,4000 -4,3000 -5,2400 -6,2000 -8,1500 -10,1200 -12,1000 -15,800 -16,750 -20,600 -24,500 -25,480 -30,400 -32,375 -40,300 -48,250 -50,240 -60,200 -75,160 -80,150 -96,125 -100,120$$

Tātaihia te tapeke mō ia takirua.

$$-1+12000=11999 -2+6000=5998 -3+4000=3997 -4+3000=2996 -5+2400=2395 -6+2000=1994 -8+1500=1492 -10+1200=1190 -12+1000=988 -15+800=785 -16+750=734 -20+600=580 -24+500=476 -25+480=455 -30+400=370 -32+375=343 -40+300=260 -48+250=202 -50+240=190 -60+200=140 -75+160=85 -80+150=70 -96+125=29 -100+120=20$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=-100 b=120$$

Me tuhi anō te kīanga whakatauwehe \left(s+a\right)\left(s+b\right) mā ngā uara i tātaihia.

$$\left(s-100\right)\left(s+120\right)$$

Hei kimi otinga whārite, me whakaoti te s-100=0 me te s+120=0.

$$s=100 s=-120$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$s^{2}+20s=12000$$

Tangohia te 12000 mai i ngā taha e rua.

$$s^{2}+20s-12000=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei s^{2}+as+bs-12000. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=20 ab=1\left(-12000\right)=-12000$$

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 -12000.

$$-1,12000 -2,6000 -3,4000 -4,3000 -5,2400 -6,2000 -8,1500 -10,1200 -12,1000 -15,800 -16,750 -20,600 -24,500 -25,480 -30,400 -32,375 -40,300 -48,250 -50,240 -60,200 -75,160 -80,150 -96,125 -100,120$$

Tātaihia te tapeke mō ia takirua.

$$-1+12000=11999 -2+6000=5998 -3+4000=3997 -4+3000=2996 -5+2400=2395 -6+2000=1994 -8+1500=1492 -10+1200=1190 -12+1000=988 -15+800=785 -16+750=734 -20+600=580 -24+500=476 -25+480=455 -30+400=370 -32+375=343 -40+300=260 -48+250=202 -50+240=190 -60+200=140 -75+160=85 -80+150=70 -96+125=29 -100+120=20$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=-100 b=120$$

Tuhia anō te s^{2}+20s-12000 hei \left(s^{2}-100s\right)+\left(120s-12000\right).

$$\left(s^{2}-100s\right)+\left(120s-12000\right)$$

Tauwehea te s i te tuatahi me te 120 i te rōpū tuarua.

$$s\left(s-100\right)+120\left(s-100\right)$$

Whakatauwehea atu te kīanga pātahi s-100 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(s-100\right)\left(s+120\right)$$

Hei kimi otinga whārite, me whakaoti te s-100=0 me te s+120=0.

$$s=100 s=-120$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$s^{2}+20s=12000$$

Tangohia te 12000 mai i ngā taha e rua.

$$s^{2}+20s-12000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 20 mō b, me -12000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 20.

$$s=\frac{-20±\sqrt{400-4\left(-12000\right)}}{2}$$

Whakareatia -4 ki te -12000.

$$s=\frac{-20±\sqrt{400+48000}}{2}$$

Tāpiri 400 ki te 48000.

$$s=\frac{-20±\sqrt{48400}}{2}$$

Tuhia te pūtakerua o te 48400.

$$s=\frac{-20±220}{2}$$

Nā, me whakaoti te whārite s=\frac{-20±220}{2} ina he tāpiri te ±. Tāpiri -20 ki te 220.

$$s=\frac{200}{2}$$

Whakawehe 200 ki te 2.

$$s=100$$

Nā, me whakaoti te whārite s=\frac{-20±220}{2} ina he tango te ±. Tango 220 mai i -20.

$$s=-\frac{240}{2}$$

Whakawehe -240 ki te 2.

$$s=-120$$

Kua oti te whārite te whakatau.

$$s=100 s=-120$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$s^{2}+20s=12000$$

Whakawehea te 20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 10. Nā, tāpiria te pūrua o te 10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$s^{2}+20s+10^{2}=12000+10^{2}$$

Pūrua 10.

$$s^{2}+20s+100=12000+100$$

Tāpiri 12000 ki te 100.

$$s^{2}+20s+100=12100$$

Tauwehea s^{2}+20s+100. 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}.

$$\left(s+10\right)^{2}=12100$$

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

$$\sqrt{\left(s+10\right)^{2}}=\sqrt{12100}$$

Whakarūnātia.

$$s+10=110 s+10=-110$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$s=100 s=-120$$