Whakaoti i te 13n^2-41n-120=0

$$13 n ^ { 2 } - 41 n - 120 = 0$$

Whakaoti mō n

$$n = -\frac{24}{13} = -1\frac{11}{13} \approx -1.846153846$$

$$n=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 13n^{2}+an+bn-120. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-41 ab=13\left(-120\right)=-1560$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -1560.

$$1,-1560 2,-780 3,-520 4,-390 5,-312 6,-260 8,-195 10,-156 12,-130 13,-120 15,-104 20,-78 24,-65 26,-60 30,-52 39,-40$$

Tātaihia te tapeke mō ia takirua.

$$1-1560=-1559 2-780=-778 3-520=-517 4-390=-386 5-312=-307 6-260=-254 8-195=-187 10-156=-146 12-130=-118 13-120=-107 15-104=-89 20-78=-58 24-65=-41 26-60=-34 30-52=-22 39-40=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -41.

$$a=-65 b=24$$

Tuhia anō te 13n^{2}-41n-120 hei \left(13n^{2}-65n\right)+\left(24n-120\right).

$$\left(13n^{2}-65n\right)+\left(24n-120\right)$$

Tauwehea te 13n i te tuatahi me te 24 i te rōpū tuarua.

$$13n\left(n-5\right)+24\left(n-5\right)$$

Whakatauwehea atu te kīanga pātahi n-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-5\right)\left(13n+24\right)$$

Hei kimi otinga whārite, me whakaoti te n-5=0 me te 13n+24=0.

$$n=5 n=-\frac{24}{13}$$

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.

$$13n^{2}-41n-120=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 13 mō a, -41 mō b, me -120 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 13\left(-120\right)}}{2\times 13}$$

Pūrua -41.

$$n=\frac{-\left(-41\right)±\sqrt{1681-4\times 13\left(-120\right)}}{2\times 13}$$

Whakareatia -4 ki te 13.

$$n=\frac{-\left(-41\right)±\sqrt{1681-52\left(-120\right)}}{2\times 13}$$

Whakareatia -52 ki te -120.

$$n=\frac{-\left(-41\right)±\sqrt{1681+6240}}{2\times 13}$$

Tāpiri 1681 ki te 6240.

$$n=\frac{-\left(-41\right)±\sqrt{7921}}{2\times 13}$$

Tuhia te pūtakerua o te 7921.

$$n=\frac{-\left(-41\right)±89}{2\times 13}$$

Ko te tauaro o -41 ko 41.

$$n=\frac{41±89}{2\times 13}$$

Whakareatia 2 ki te 13.

$$n=\frac{41±89}{26}$$

Nā, me whakaoti te whārite n=\frac{41±89}{26} ina he tāpiri te ±. Tāpiri 41 ki te 89.

$$n=\frac{130}{26}$$

Whakawehe 130 ki te 26.

$$n=5$$

Nā, me whakaoti te whārite n=\frac{41±89}{26} ina he tango te ±. Tango 89 mai i 41.

$$n=-\frac{48}{26}$$

Whakahekea te hautanga \frac{-48}{26} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n=-\frac{24}{13}$$

Kua oti te whārite te whakatau.

$$n=5 n=-\frac{24}{13}$$

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.

$$13n^{2}-41n-120=0$$

Me tāpiri 120 ki ngā taha e rua o te whārite.

$$13n^{2}-41n-120-\left(-120\right)=-\left(-120\right)$$

Mā te tango i te -120 i a ia ake anō ka toe ko te 0.

$$13n^{2}-41n=-\left(-120\right)$$

Tango -120 mai i 0.

$$13n^{2}-41n=120$$

Whakawehea ngā taha e rua ki te 13.

$$\frac{13n^{2}-41n}{13}=\frac{120}{13}$$

Mā te whakawehe ki te 13 ka wetekia te whakareanga ki te 13.

$$n^{2}-\frac{41}{13}n=\frac{120}{13}$$

Whakawehea te -\frac{41}{13}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{41}{26}. Nā, tāpiria te pūrua o te -\frac{41}{26} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{41}{13}n+\left(-\frac{41}{26}\right)^{2}=\frac{120}{13}+\left(-\frac{41}{26}\right)^{2}$$

Pūruatia -\frac{41}{26} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{41}{13}n+\frac{1681}{676}=\frac{120}{13}+\frac{1681}{676}$$

Tāpiri \frac{120}{13} ki te \frac{1681}{676} 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.

$$n^{2}-\frac{41}{13}n+\frac{1681}{676}=\frac{7921}{676}$$

Tauwehea n^{2}-\frac{41}{13}n+\frac{1681}{676}. 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}.

$$\left(n-\frac{41}{26}\right)^{2}=\frac{7921}{676}$$

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

$$\sqrt{\left(n-\frac{41}{26}\right)^{2}}=\sqrt{\frac{7921}{676}}$$

Whakarūnātia.

$$n-\frac{41}{26}=\frac{89}{26} n-\frac{41}{26}=-\frac{89}{26}$$

Me tāpiri \frac{41}{26} ki ngā taha e rua o te whārite.

$$n=5 n=-\frac{24}{13}$$