Whakaoti i te n^2-n=240

$$n ^ { 2 } - n = 240$$

Whakaoti mō n

$$n=-15$$

$$n=16$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 240 mai i ngā taha e rua.

$$n^{2}-n-240=0$$

Hei whakaoti i te whārite, whakatauwehea te n^{2}-n-240 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.

$$a+b=-1 ab=-240$$

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 -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Tātaihia te tapeke mō ia takirua.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

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

$$a=-16 b=15$$

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

$$\left(n-16\right)\left(n+15\right)$$

Hei kimi otinga whārite, me whakaoti te n-16=0 me te n+15=0.

$$n=16 n=-15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 240 mai i ngā taha e rua.

$$n^{2}-n-240=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei n^{2}+an+bn-240. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-240\right)=-240$$

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 -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Tātaihia te tapeke mō ia takirua.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

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

$$a=-16 b=15$$

Tuhia anō te n^{2}-n-240 hei \left(n^{2}-16n\right)+\left(15n-240\right).

$$\left(n^{2}-16n\right)+\left(15n-240\right)$$

Tauwehea te n i te tuatahi me te 15 i te rōpū tuarua.

$$n\left(n-16\right)+15\left(n-16\right)$$

Whakatauwehea atu te kīanga pātahi n-16 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-16\right)\left(n+15\right)$$

Hei kimi otinga whārite, me whakaoti te n-16=0 me te n+15=0.

$$n=16 n=-15$$

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.

$$n^{2}-n=240$$

Me tango 240 mai i ngā taha e rua o te whārite.

$$n^{2}-n-240=240-240$$

Mā te tango i te 240 i a ia ake anō ka toe ko te 0.

$$n^{2}-n-240=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -240 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\left(-240\right)}}{2}$$

Whakareatia -4 ki te -240.

$$n=\frac{-\left(-1\right)±\sqrt{1+960}}{2}$$

Tāpiri 1 ki te 960.

$$n=\frac{-\left(-1\right)±\sqrt{961}}{2}$$

Tuhia te pūtakerua o te 961.

$$n=\frac{-\left(-1\right)±31}{2}$$

Ko te tauaro o -1 ko 1.

$$n=\frac{1±31}{2}$$

Nā, me whakaoti te whārite n=\frac{1±31}{2} ina he tāpiri te ±. Tāpiri 1 ki te 31.

$$n=\frac{32}{2}$$

Whakawehe 32 ki te 2.

$$n=16$$

Nā, me whakaoti te whārite n=\frac{1±31}{2} ina he tango te ±. Tango 31 mai i 1.

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

Whakawehe -30 ki te 2.

$$n=-15$$

Kua oti te whārite te whakatau.

$$n=16 n=-15$$

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.

$$n^{2}-n=240$$

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.

$$n^{2}-n+\left(-\frac{1}{2}\right)^{2}=240+\left(-\frac{1}{2}\right)^{2}$$

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

$$n^{2}-n+\frac{1}{4}=240+\frac{1}{4}$$

Tāpiri 240 ki te \frac{1}{4}.

$$n^{2}-n+\frac{1}{4}=\frac{961}{4}$$

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}.

$$\left(n-\frac{1}{2}\right)^{2}=\frac{961}{4}$$

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

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{961}{4}}$$

Whakarūnātia.

$$n-\frac{1}{2}=\frac{31}{2} n-\frac{1}{2}=-\frac{31}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$n=16 n=-15$$