Whakaoti i te 120x^2+8000x-92000=0

$$120 { x }^{ 2 } +8000x-92000=0$$

Whakaoti mō x

$$x = -\frac{230}{3} = -76\frac{2}{3} \approx -76.666666667$$

$$x=10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 40.

$$3x^{2}+200x-2300=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-2300. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=200 ab=3\left(-2300\right)=-6900$$

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 -6900.

$$-1,6900 -2,3450 -3,2300 -4,1725 -5,1380 -6,1150 -10,690 -12,575 -15,460 -20,345 -23,300 -25,276 -30,230 -46,150 -50,138 -60,115 -69,100 -75,92$$

Tātaihia te tapeke mō ia takirua.

$$-1+6900=6899 -2+3450=3448 -3+2300=2297 -4+1725=1721 -5+1380=1375 -6+1150=1144 -10+690=680 -12+575=563 -15+460=445 -20+345=325 -23+300=277 -25+276=251 -30+230=200 -46+150=104 -50+138=88 -60+115=55 -69+100=31 -75+92=17$$

Ko te otinga te takirua ka hoatu i te tapeke 200.

$$a=-30 b=230$$

Tuhia anō te 3x^{2}+200x-2300 hei \left(3x^{2}-30x\right)+\left(230x-2300\right).

$$\left(3x^{2}-30x\right)+\left(230x-2300\right)$$

Tauwehea te 3x i te tuatahi me te 230 i te rōpū tuarua.

$$3x\left(x-10\right)+230\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(3x+230\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te 3x+230=0.

$$x=10 x=-\frac{230}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$120x^{2}+8000x-92000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 120 mō a, 8000 mō b, me -92000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8000±\sqrt{8000^{2}-4\times 120\left(-92000\right)}}{2\times 120}$$

Pūrua 8000.

$$x=\frac{-8000±\sqrt{64000000-4\times 120\left(-92000\right)}}{2\times 120}$$

Whakareatia -4 ki te 120.

$$x=\frac{-8000±\sqrt{64000000-480\left(-92000\right)}}{2\times 120}$$

Whakareatia -480 ki te -92000.

$$x=\frac{-8000±\sqrt{64000000+44160000}}{2\times 120}$$

Tāpiri 64000000 ki te 44160000.

$$x=\frac{-8000±\sqrt{108160000}}{2\times 120}$$

Tuhia te pūtakerua o te 108160000.

$$x=\frac{-8000±10400}{2\times 120}$$

Whakareatia 2 ki te 120.

$$x=\frac{-8000±10400}{240}$$

Nā, me whakaoti te whārite x=\frac{-8000±10400}{240} ina he tāpiri te ±. Tāpiri -8000 ki te 10400.

$$x=\frac{2400}{240}$$

Whakawehe 2400 ki te 240.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{-8000±10400}{240} ina he tango te ±. Tango 10400 mai i -8000.

$$x=-\frac{18400}{240}$$

Whakahekea te hautanga \frac{-18400}{240} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 80.

$$x=-\frac{230}{3}$$

Kua oti te whārite te whakatau.

$$x=10 x=-\frac{230}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$120x^{2}+8000x-92000=0$$

Me tāpiri 92000 ki ngā taha e rua o te whārite.

$$120x^{2}+8000x-92000-\left(-92000\right)=-\left(-92000\right)$$

Mā te tango i te -92000 i a ia ake anō ka toe ko te 0.

$$120x^{2}+8000x=-\left(-92000\right)$$

Tango -92000 mai i 0.

$$120x^{2}+8000x=92000$$

Whakawehea ngā taha e rua ki te 120.

$$\frac{120x^{2}+8000x}{120}=\frac{92000}{120}$$

Mā te whakawehe ki te 120 ka wetekia te whakareanga ki te 120.

$$x^{2}+\frac{8000}{120}x=\frac{92000}{120}$$

Whakahekea te hautanga \frac{8000}{120} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 40.

$$x^{2}+\frac{200}{3}x=\frac{92000}{120}$$

Whakahekea te hautanga \frac{92000}{120} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 40.

$$x^{2}+\frac{200}{3}x=\frac{2300}{3}$$

Whakawehea te \frac{200}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{100}{3}. Nā, tāpiria te pūrua o te \frac{100}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{200}{3}x+\left(\frac{100}{3}\right)^{2}=\frac{2300}{3}+\left(\frac{100}{3}\right)^{2}$$

Pūruatia \frac{100}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{2300}{3}+\frac{10000}{9}$$

Tāpiri \frac{2300}{3} ki te \frac{10000}{9} 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.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{16900}{9}$$

Tauwehea x^{2}+\frac{200}{3}x+\frac{10000}{9}. 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}.

$$\left(x+\frac{100}{3}\right)^{2}=\frac{16900}{9}$$

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

$$\sqrt{\left(x+\frac{100}{3}\right)^{2}}=\sqrt{\frac{16900}{9}}$$

Whakarūnātia.

$$x+\frac{100}{3}=\frac{130}{3} x+\frac{100}{3}=-\frac{130}{3}$$

Me tango \frac{100}{3} mai i ngā taha e rua o te whārite.

$$x=10 x=-\frac{230}{3}$$