Whakaoti i te n^2-7n=8

$$n ^ { 2 } - 7 n = 8$$

Whakaoti mō n

$$n=-1$$

$$n=8$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 8 mai i ngā taha e rua.

$$n^{2}-7n-8=0$$

Hei whakaoti i te whārite, whakatauwehea te n^{2}-7n-8 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.

$$a+b=-7 ab=-8$$

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 -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-8 b=1$$

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

$$\left(n-8\right)\left(n+1\right)$$

Hei kimi otinga whārite, me whakaoti te n-8=0 me te n+1=0.

$$n=8 n=-1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 8 mai i ngā taha e rua.

$$n^{2}-7n-8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei n^{2}+an+bn-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=1\left(-8\right)=-8$$

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 -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-8 b=1$$

Tuhia anō te n^{2}-7n-8 hei \left(n^{2}-8n\right)+\left(n-8\right).

$$\left(n^{2}-8n\right)+\left(n-8\right)$$

Whakatauwehea atu n i te n^{2}-8n.

$$n\left(n-8\right)+n-8$$

Whakatauwehea atu te kīanga pātahi n-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-8\right)\left(n+1\right)$$

Hei kimi otinga whārite, me whakaoti te n-8=0 me te n+1=0.

$$n=8 n=-1$$

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.

$$n^{2}-7n=8$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$n^{2}-7n-8=8-8$$

Mā te tango i te 8 i a ia ake anō ka toe ko te 0.

$$n^{2}-7n-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -7 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-8\right)}}{2}$$

Pūrua -7.

$$n=\frac{-\left(-7\right)±\sqrt{49-4\left(-8\right)}}{2}$$

Whakareatia -4 ki te -8.

$$n=\frac{-\left(-7\right)±\sqrt{49+32}}{2}$$

Tāpiri 49 ki te 32.

$$n=\frac{-\left(-7\right)±\sqrt{81}}{2}$$

Tuhia te pūtakerua o te 81.

$$n=\frac{-\left(-7\right)±9}{2}$$

Ko te tauaro o -7 ko 7.

$$n=\frac{7±9}{2}$$

Nā, me whakaoti te whārite n=\frac{7±9}{2} ina he tāpiri te ±. Tāpiri 7 ki te 9.

$$n=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$n=8$$

Nā, me whakaoti te whārite n=\frac{7±9}{2} ina he tango te ±. Tango 9 mai i 7.

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

Whakawehe -2 ki te 2.

$$n=-1$$

Kua oti te whārite te whakatau.

$$n=8 n=-1$$

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.

$$n^{2}-7n=8$$

Whakawehea te -7, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{2}. Nā, tāpiria te pūrua o te -\frac{7}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-7n+\left(-\frac{7}{2}\right)^{2}=8+\left(-\frac{7}{2}\right)^{2}$$

Pūruatia -\frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-7n+\frac{49}{4}=8+\frac{49}{4}$$

Tāpiri 8 ki te \frac{49}{4}.

$$n^{2}-7n+\frac{49}{4}=\frac{81}{4}$$

Tauwehea n^{2}-7n+\frac{49}{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}.

$$\left(n-\frac{7}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(n-\frac{7}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

$$n-\frac{7}{2}=\frac{9}{2} n-\frac{7}{2}=-\frac{9}{2}$$

Me tāpiri \frac{7}{2} ki ngā taha e rua o te whārite.

$$n=8 n=-1$$