Whakaoti i te 2k^2+9k=-7

$$2 k ^ { 2 } + 9 k = - 7$$

Whakaoti mō k

$$k = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$k=-1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 7 ki ngā taha e rua.

$$2k^{2}+9k+7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2k^{2}+ak+bk+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=2\times 7=14$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 14.

$$1,14 2,7$$

Tātaihia te tapeke mō ia takirua.

$$1+14=15 2+7=9$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=2 b=7$$

Tuhia anō te 2k^{2}+9k+7 hei \left(2k^{2}+2k\right)+\left(7k+7\right).

$$\left(2k^{2}+2k\right)+\left(7k+7\right)$$

Tauwehea te 2k i te tuatahi me te 7 i te rōpū tuarua.

$$2k\left(k+1\right)+7\left(k+1\right)$$

Whakatauwehea atu te kīanga pātahi k+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(k+1\right)\left(2k+7\right)$$

Hei kimi otinga whārite, me whakaoti te k+1=0 me te 2k+7=0.

$$k=-1 k=-\frac{7}{2}$$

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.

$$2k^{2}+9k=-7$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$2k^{2}+9k-\left(-7\right)=-7-\left(-7\right)$$

Mā te tango i te -7 i a ia ake anō ka toe ko te 0.

$$2k^{2}+9k-\left(-7\right)=0$$

Tango -7 mai i 0.

$$2k^{2}+9k+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 9 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-9±\sqrt{9^{2}-4\times 2\times 7}}{2\times 2}$$

Pūrua 9.

$$k=\frac{-9±\sqrt{81-4\times 2\times 7}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$k=\frac{-9±\sqrt{81-8\times 7}}{2\times 2}$$

Whakareatia -8 ki te 7.

$$k=\frac{-9±\sqrt{81-56}}{2\times 2}$$

Tāpiri 81 ki te -56.

$$k=\frac{-9±\sqrt{25}}{2\times 2}$$

Tuhia te pūtakerua o te 25.

$$k=\frac{-9±5}{2\times 2}$$

Whakareatia 2 ki te 2.

$$k=\frac{-9±5}{4}$$

Nā, me whakaoti te whārite k=\frac{-9±5}{4} ina he tāpiri te ±. Tāpiri -9 ki te 5.

$$k=-\frac{4}{4}$$

Whakawehe -4 ki te 4.

$$k=-1$$

Nā, me whakaoti te whārite k=\frac{-9±5}{4} ina he tango te ±. Tango 5 mai i -9.

$$k=-\frac{14}{4}$$

Whakahekea te hautanga \frac{-14}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$k=-\frac{7}{2}$$

Kua oti te whārite te whakatau.

$$k=-1 k=-\frac{7}{2}$$

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.

$$2k^{2}+9k=-7$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2k^{2}+9k}{2}=-\frac{7}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$k^{2}+\frac{9}{2}k=-\frac{7}{2}$$

Whakawehea te \frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{4}. Nā, tāpiria te pūrua o te \frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}+\frac{9}{2}k+\left(\frac{9}{4}\right)^{2}=-\frac{7}{2}+\left(\frac{9}{4}\right)^{2}$$

Pūruatia \frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$k^{2}+\frac{9}{2}k+\frac{81}{16}=-\frac{7}{2}+\frac{81}{16}$$

Tāpiri -\frac{7}{2} ki te \frac{81}{16} 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.

$$k^{2}+\frac{9}{2}k+\frac{81}{16}=\frac{25}{16}$$

Tauwehea te k^{2}+\frac{9}{2}k+\frac{81}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(k+\frac{9}{4}\right)^{2}=\frac{25}{16}$$

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

$$\sqrt{\left(k+\frac{9}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Whakarūnātia.

$$k+\frac{9}{4}=\frac{5}{4} k+\frac{9}{4}=-\frac{5}{4}$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

$$k=-1 k=-\frac{7}{2}$$