Whakaoti i te n^2-11n-60=0

$$n ^ { 2 } - 11 n - 60 = 0$$

Whakaoti mō n

$$n=-4$$

$$n=15$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te n^{2}-11n-60 mā te whakamahi i te tātai n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-15 b=4$$

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

$$\left(n-15\right)\left(n+4\right)$$

Hei kimi otinga whārite, me whakaoti te n-15=0 me te n+4=0.

$$n=15 n=-4$$

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 n^{2}+an+bn-60. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=1\left(-60\right)=-60$$

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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-15 b=4$$

Tuhia anō te n^{2}-11n-60 hei \left(n^{2}-15n\right)+\left(4n-60\right).

$$\left(n^{2}-15n\right)+\left(4n-60\right)$$

Tauwehea te n i te tuatahi me te 4 i te rōpū tuarua.

$$n\left(n-15\right)+4\left(n-15\right)$$

Whakatauwehea atu te kīanga pātahi n-15 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-15\right)\left(n+4\right)$$

Hei kimi otinga whārite, me whakaoti te n-15=0 me te n+4=0.

$$n=15 n=-4$$

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.

$$n^{2}-11n-60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -11 mō b, me -60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-60\right)}}{2}$$

Pūrua -11.

$$n=\frac{-\left(-11\right)±\sqrt{121-4\left(-60\right)}}{2}$$

Whakareatia -4 ki te -60.

$$n=\frac{-\left(-11\right)±\sqrt{121+240}}{2}$$

Tāpiri 121 ki te 240.

$$n=\frac{-\left(-11\right)±\sqrt{361}}{2}$$

Tuhia te pūtakerua o te 361.

$$n=\frac{-\left(-11\right)±19}{2}$$

Ko te tauaro o -11 ko 11.

$$n=\frac{11±19}{2}$$

Nā, me whakaoti te whārite n=\frac{11±19}{2} ina he tāpiri te ±. Tāpiri 11 ki te 19.

$$n=\frac{30}{2}$$

Whakawehe 30 ki te 2.

$$n=15$$

Nā, me whakaoti te whārite n=\frac{11±19}{2} ina he tango te ±. Tango 19 mai i 11.

$$n=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$n=-4$$

Kua oti te whārite te whakatau.

$$n=15 n=-4$$

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.

$$n^{2}-11n-60=0$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$n^{2}-11n-60-\left(-60\right)=-\left(-60\right)$$

Mā te tango i te -60 i a ia ake anō ka toe ko te 0.

$$n^{2}-11n=-\left(-60\right)$$

Tango -60 mai i 0.

$$n^{2}-11n=60$$

Whakawehea te -11, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{2}. Nā, tāpiria te pūrua o te -\frac{11}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-11n+\left(-\frac{11}{2}\right)^{2}=60+\left(-\frac{11}{2}\right)^{2}$$

Pūruatia -\frac{11}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-11n+\frac{121}{4}=60+\frac{121}{4}$$

Tāpiri 60 ki te \frac{121}{4}.

$$n^{2}-11n+\frac{121}{4}=\frac{361}{4}$$

Tauwehea n^{2}-11n+\frac{121}{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}.

$$\left(n-\frac{11}{2}\right)^{2}=\frac{361}{4}$$

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

$$\sqrt{\left(n-\frac{11}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Whakarūnātia.

$$n-\frac{11}{2}=\frac{19}{2} n-\frac{11}{2}=-\frac{19}{2}$$

Me tāpiri \frac{11}{2} ki ngā taha e rua o te whārite.

$$n=15 n=-4$$