Whakaoti i te 7n^2+39n-18=0

$$7 n ^ { 2 } + 39 n - 18 = 0$$

Whakaoti mō n

$$n=-6$$

$$n=\frac{3}{7}\approx 0.428571429$$

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 7n^{2}+an+bn-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=39 ab=7\left(-18\right)=-126$$

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 -126.

$$-1,126 -2,63 -3,42 -6,21 -7,18 -9,14$$

Tātaihia te tapeke mō ia takirua.

$$-1+126=125 -2+63=61 -3+42=39 -6+21=15 -7+18=11 -9+14=5$$

Ko te otinga te takirua ka hoatu i te tapeke 39.

$$a=-3 b=42$$

Tuhia anō te 7n^{2}+39n-18 hei \left(7n^{2}-3n\right)+\left(42n-18\right).

$$\left(7n^{2}-3n\right)+\left(42n-18\right)$$

Tauwehea te n i te tuatahi me te 6 i te rōpū tuarua.

$$n\left(7n-3\right)+6\left(7n-3\right)$$

Whakatauwehea atu te kīanga pātahi 7n-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7n-3\right)\left(n+6\right)$$

Hei kimi otinga whārite, me whakaoti te 7n-3=0 me te n+6=0.

$$n=\frac{3}{7} n=-6$$

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.

$$7n^{2}+39n-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 39 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-39±\sqrt{39^{2}-4\times 7\left(-18\right)}}{2\times 7}$$

Pūrua 39.

$$n=\frac{-39±\sqrt{1521-4\times 7\left(-18\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$n=\frac{-39±\sqrt{1521-28\left(-18\right)}}{2\times 7}$$

Whakareatia -28 ki te -18.

$$n=\frac{-39±\sqrt{1521+504}}{2\times 7}$$

Tāpiri 1521 ki te 504.

$$n=\frac{-39±\sqrt{2025}}{2\times 7}$$

Tuhia te pūtakerua o te 2025.

$$n=\frac{-39±45}{2\times 7}$$

Whakareatia 2 ki te 7.

$$n=\frac{-39±45}{14}$$

Nā, me whakaoti te whārite n=\frac{-39±45}{14} ina he tāpiri te ±. Tāpiri -39 ki te 45.

$$n=\frac{6}{14}$$

Whakahekea te hautanga \frac{6}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n=\frac{3}{7}$$

Nā, me whakaoti te whārite n=\frac{-39±45}{14} ina he tango te ±. Tango 45 mai i -39.

$$n=-\frac{84}{14}$$

Whakawehe -84 ki te 14.

$$n=-6$$

Kua oti te whārite te whakatau.

$$n=\frac{3}{7} n=-6$$

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.

$$7n^{2}+39n-18=0$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$7n^{2}+39n-18-\left(-18\right)=-\left(-18\right)$$

Mā te tango i te -18 i a ia ake anō ka toe ko te 0.

$$7n^{2}+39n=-\left(-18\right)$$

Tango -18 mai i 0.

$$7n^{2}+39n=18$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7n^{2}+39n}{7}=\frac{18}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$n^{2}+\frac{39}{7}n=\frac{18}{7}$$

Whakawehea te \frac{39}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{39}{14}. Nā, tāpiria te pūrua o te \frac{39}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+\frac{39}{7}n+\left(\frac{39}{14}\right)^{2}=\frac{18}{7}+\left(\frac{39}{14}\right)^{2}$$

Pūruatia \frac{39}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+\frac{39}{7}n+\frac{1521}{196}=\frac{18}{7}+\frac{1521}{196}$$

Tāpiri \frac{18}{7} ki te \frac{1521}{196} 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.

$$n^{2}+\frac{39}{7}n+\frac{1521}{196}=\frac{2025}{196}$$

Tauwehea n^{2}+\frac{39}{7}n+\frac{1521}{196}. 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}.

$$\left(n+\frac{39}{14}\right)^{2}=\frac{2025}{196}$$

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

$$\sqrt{\left(n+\frac{39}{14}\right)^{2}}=\sqrt{\frac{2025}{196}}$$

Whakarūnātia.

$$n+\frac{39}{14}=\frac{45}{14} n+\frac{39}{14}=-\frac{45}{14}$$

Me tango \frac{39}{14} mai i ngā taha e rua o te whārite.

$$n=\frac{3}{7} n=-6$$