Whakaoti i te 28k^2+k-2=0

$$28 k ^ { 2 } + k - 2 = 0$$

Whakaoti mō k

$$k=\frac{1}{4}=0.25$$

$$k=-\frac{2}{7}\approx -0.285714286$$

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 28k^{2}+ak+bk-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=28\left(-2\right)=-56$$

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 -56.

$$-1,56 -2,28 -4,14 -7,8$$

Tātaihia te tapeke mō ia takirua.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

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

$$a=-7 b=8$$

Tuhia anō te 28k^{2}+k-2 hei \left(28k^{2}-7k\right)+\left(8k-2\right).

$$\left(28k^{2}-7k\right)+\left(8k-2\right)$$

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

$$7k\left(4k-1\right)+2\left(4k-1\right)$$

Whakatauwehea atu te kīanga pātahi 4k-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4k-1\right)\left(7k+2\right)$$

Hei kimi otinga whārite, me whakaoti te 4k-1=0 me te 7k+2=0.

$$k=\frac{1}{4} k=-\frac{2}{7}$$

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.

$$28k^{2}+k-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 28 mō a, 1 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-1±\sqrt{1^{2}-4\times 28\left(-2\right)}}{2\times 28}$$

Pūrua 1.

$$k=\frac{-1±\sqrt{1-4\times 28\left(-2\right)}}{2\times 28}$$

Whakareatia -4 ki te 28.

$$k=\frac{-1±\sqrt{1-112\left(-2\right)}}{2\times 28}$$

Whakareatia -112 ki te -2.

$$k=\frac{-1±\sqrt{1+224}}{2\times 28}$$

Tāpiri 1 ki te 224.

$$k=\frac{-1±\sqrt{225}}{2\times 28}$$

Tuhia te pūtakerua o te 225.

$$k=\frac{-1±15}{2\times 28}$$

Whakareatia 2 ki te 28.

$$k=\frac{-1±15}{56}$$

Nā, me whakaoti te whārite k=\frac{-1±15}{56} ina he tāpiri te ±. Tāpiri -1 ki te 15.

$$k=\frac{14}{56}$$

Whakahekea te hautanga \frac{14}{56} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$k=\frac{1}{4}$$

Nā, me whakaoti te whārite k=\frac{-1±15}{56} ina he tango te ±. Tango 15 mai i -1.

$$k=-\frac{16}{56}$$

Whakahekea te hautanga \frac{-16}{56} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Kua oti te whārite te whakatau.

$$k=\frac{1}{4} k=-\frac{2}{7}$$

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.

$$28k^{2}+k-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$28k^{2}+k-2-\left(-2\right)=-\left(-2\right)$$

Mā te tango i te -2 i a ia ake anō ka toe ko te 0.

$$28k^{2}+k=-\left(-2\right)$$

Tango -2 mai i 0.

$$28k^{2}+k=2$$

Whakawehea ngā taha e rua ki te 28.

$$\frac{28k^{2}+k}{28}=\frac{2}{28}$$

Mā te whakawehe ki te 28 ka wetekia te whakareanga ki te 28.

$$k^{2}+\frac{1}{28}k=\frac{2}{28}$$

Whakahekea te hautanga \frac{2}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$k^{2}+\frac{1}{28}k=\frac{1}{14}$$

Whakawehea te \frac{1}{28}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{56}. Nā, tāpiria te pūrua o te \frac{1}{56} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$k^{2}+\frac{1}{28}k+\left(\frac{1}{56}\right)^{2}=\frac{1}{14}+\left(\frac{1}{56}\right)^{2}$$

Pūruatia \frac{1}{56} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$k^{2}+\frac{1}{28}k+\frac{1}{3136}=\frac{1}{14}+\frac{1}{3136}$$

Tāpiri \frac{1}{14} ki te \frac{1}{3136} 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.

$$k^{2}+\frac{1}{28}k+\frac{1}{3136}=\frac{225}{3136}$$

Tauwehea k^{2}+\frac{1}{28}k+\frac{1}{3136}. 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}.

$$\left(k+\frac{1}{56}\right)^{2}=\frac{225}{3136}$$

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

$$\sqrt{\left(k+\frac{1}{56}\right)^{2}}=\sqrt{\frac{225}{3136}}$$

Whakarūnātia.

$$k+\frac{1}{56}=\frac{15}{56} k+\frac{1}{56}=-\frac{15}{56}$$

Me tango \frac{1}{56} mai i ngā taha e rua o te whārite.

$$k=\frac{1}{4} k=-\frac{2}{7}$$