Whakaoti i te 0=5n^2+1205n-90300

$$0 = 5 n ^ { 2 } + 1205 n - 90300$$

Whakaoti mō n

$$n=-301$$

$$n=60$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$5n^{2}+1205n-90300=0$$

Whakawehea ngā taha e rua ki te 5.

$$n^{2}+241n-18060=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei n^{2}+an+bn-18060. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=241 ab=1\left(-18060\right)=-18060$$

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 -18060.

$$-1,18060 -2,9030 -3,6020 -4,4515 -5,3612 -6,3010 -7,2580 -10,1806 -12,1505 -14,1290 -15,1204 -20,903 -21,860 -28,645 -30,602 -35,516 -42,430 -43,420 -60,301 -70,258 -84,215 -86,210 -105,172 -129,140$$

Tātaihia te tapeke mō ia takirua.

$$-1+18060=18059 -2+9030=9028 -3+6020=6017 -4+4515=4511 -5+3612=3607 -6+3010=3004 -7+2580=2573 -10+1806=1796 -12+1505=1493 -14+1290=1276 -15+1204=1189 -20+903=883 -21+860=839 -28+645=617 -30+602=572 -35+516=481 -42+430=388 -43+420=377 -60+301=241 -70+258=188 -84+215=131 -86+210=124 -105+172=67 -129+140=11$$

Ko te otinga te takirua ka hoatu i te tapeke 241.

$$a=-60 b=301$$

Tuhia anō te n^{2}+241n-18060 hei \left(n^{2}-60n\right)+\left(301n-18060\right).

$$\left(n^{2}-60n\right)+\left(301n-18060\right)$$

Tauwehea te n i te tuatahi me te 301 i te rōpū tuarua.

$$n\left(n-60\right)+301\left(n-60\right)$$

Whakatauwehea atu te kīanga pātahi n-60 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-60\right)\left(n+301\right)$$

Hei kimi otinga whārite, me whakaoti te n-60=0 me te n+301=0.

$$n=60 n=-301$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$5n^{2}+1205n-90300=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 1205 mō b, me -90300 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1205±\sqrt{1205^{2}-4\times 5\left(-90300\right)}}{2\times 5}$$

Pūrua 1205.

$$n=\frac{-1205±\sqrt{1452025-4\times 5\left(-90300\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$n=\frac{-1205±\sqrt{1452025-20\left(-90300\right)}}{2\times 5}$$

Whakareatia -20 ki te -90300.

$$n=\frac{-1205±\sqrt{1452025+1806000}}{2\times 5}$$

Tāpiri 1452025 ki te 1806000.

$$n=\frac{-1205±\sqrt{3258025}}{2\times 5}$$

Tuhia te pūtakerua o te 3258025.

$$n=\frac{-1205±1805}{2\times 5}$$

Whakareatia 2 ki te 5.

$$n=\frac{-1205±1805}{10}$$

Nā, me whakaoti te whārite n=\frac{-1205±1805}{10} ina he tāpiri te ±. Tāpiri -1205 ki te 1805.

$$n=\frac{600}{10}$$

Whakawehe 600 ki te 10.

$$n=60$$

Nā, me whakaoti te whārite n=\frac{-1205±1805}{10} ina he tango te ±. Tango 1805 mai i -1205.

$$n=-\frac{3010}{10}$$

Whakawehe -3010 ki te 10.

$$n=-301$$

Kua oti te whārite te whakatau.

$$n=60 n=-301$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$5n^{2}+1205n-90300=0$$

Me tāpiri te 90300 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$5n^{2}+1205n=90300$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5n^{2}+1205n}{5}=\frac{90300}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$n^{2}+\frac{1205}{5}n=\frac{90300}{5}$$

Whakawehe 1205 ki te 5.

$$n^{2}+241n=\frac{90300}{5}$$

Whakawehe 90300 ki te 5.

$$n^{2}+241n=18060$$

Whakawehea te 241, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{241}{2}. Nā, tāpiria te pūrua o te \frac{241}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+241n+\left(\frac{241}{2}\right)^{2}=18060+\left(\frac{241}{2}\right)^{2}$$

Pūruatia \frac{241}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+241n+\frac{58081}{4}=18060+\frac{58081}{4}$$

Tāpiri 18060 ki te \frac{58081}{4}.

$$n^{2}+241n+\frac{58081}{4}=\frac{130321}{4}$$

Tauwehea n^{2}+241n+\frac{58081}{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}.

$$\left(n+\frac{241}{2}\right)^{2}=\frac{130321}{4}$$

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

$$\sqrt{\left(n+\frac{241}{2}\right)^{2}}=\sqrt{\frac{130321}{4}}$$

Whakarūnātia.

$$n+\frac{241}{2}=\frac{361}{2} n+\frac{241}{2}=-\frac{361}{2}$$

Me tango \frac{241}{2} mai i ngā taha e rua o te whārite.

$$n=60 n=-301$$