Whakaoti i te n^2+n-110=0

$$n ^ { 2 } + n - 110 = 0$$

Whakaoti mō n

$$n=-11$$

$$n=10$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te n^{2}+n-110 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.

$$a+b=1 ab=-110$$

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 -110.

$$-1,110 -2,55 -5,22 -10,11$$

Tātaihia te tapeke mō ia takirua.

$$-1+110=109 -2+55=53 -5+22=17 -10+11=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-10 b=11$$

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

$$\left(n-10\right)\left(n+11\right)$$

Hei kimi otinga whārite, me whakaoti te n-10=0 me te n+11=0.

$$n=10 n=-11$$

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-110. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=1\left(-110\right)=-110$$

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 -110.

$$-1,110 -2,55 -5,22 -10,11$$

Tātaihia te tapeke mō ia takirua.

$$-1+110=109 -2+55=53 -5+22=17 -10+11=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-10 b=11$$

Tuhia anō te n^{2}+n-110 hei \left(n^{2}-10n\right)+\left(11n-110\right).

$$\left(n^{2}-10n\right)+\left(11n-110\right)$$

Tauwehea te n i te tuatahi me te 11 i te rōpū tuarua.

$$n\left(n-10\right)+11\left(n-10\right)$$

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

$$\left(n-10\right)\left(n+11\right)$$

Hei kimi otinga whārite, me whakaoti te n-10=0 me te n+11=0.

$$n=10 n=-11$$

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.

$$n^{2}+n-110=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 1 mō b, me -110 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-110\right)}}{2}$$

Pūrua 1.

$$n=\frac{-1±\sqrt{1-4\left(-110\right)}}{2}$$

Whakareatia -4 ki te -110.

$$n=\frac{-1±\sqrt{1+440}}{2}$$

Tāpiri 1 ki te 440.

$$n=\frac{-1±\sqrt{441}}{2}$$

Tuhia te pūtakerua o te 441.

$$n=\frac{-1±21}{2}$$

Nā, me whakaoti te whārite n=\frac{-1±21}{2} ina he tāpiri te ±. Tāpiri -1 ki te 21.

$$n=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$n=10$$

Nā, me whakaoti te whārite n=\frac{-1±21}{2} ina he tango te ±. Tango 21 mai i -1.

$$n=-\frac{22}{2}$$

Whakawehe -22 ki te 2.

$$n=-11$$

Kua oti te whārite te whakatau.

$$n=10 n=-11$$

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.

$$n^{2}+n-110=0$$

Me tāpiri 110 ki ngā taha e rua o te whārite.

$$n^{2}+n-110-\left(-110\right)=-\left(-110\right)$$

Mā te tango i te -110 i a ia ake anō ka toe ko te 0.

$$n^{2}+n=-\left(-110\right)$$

Tango -110 mai i 0.

$$n^{2}+n=110$$

Whakawehea te 1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{2}. Nā, tāpiria te pūrua o te \frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=110+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+n+\frac{1}{4}=110+\frac{1}{4}$$

Tāpiri 110 ki te \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{441}{4}$$

Tauwehea n^{2}+n+\frac{1}{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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{441}{4}$$

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

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{441}{4}}$$

Whakarūnātia.

$$n+\frac{1}{2}=\frac{21}{2} n+\frac{1}{2}=-\frac{21}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$n=10 n=-11$$