Whakaoti i te 28k^2+k+1=0

$$28 k ^ { 2 } + k + 1 = 0$$

Whakaoti mō k

$$k=\frac{-1+\sqrt{111}i}{56}\approx -0.017857143+0.188136674i$$

$$k=\frac{-\sqrt{111}i-1}{56}\approx -0.017857143-0.188136674i$$

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.

$$28k^{2}+k+1=0$$

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

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

Pūrua 1.

$$k=\frac{-1±\sqrt{1-4\times 28}}{2\times 28}$$

Whakareatia -4 ki te 28.

$$k=\frac{-1±\sqrt{1-112}}{2\times 28}$$

Tāpiri 1 ki te -112.

$$k=\frac{-1±\sqrt{-111}}{2\times 28}$$

Tuhia te pūtakerua o te -111.

$$k=\frac{-1±\sqrt{111}i}{2\times 28}$$

Whakareatia 2 ki te 28.

$$k=\frac{-1±\sqrt{111}i}{56}$$

Nā, me whakaoti te whārite k=\frac{-1±\sqrt{111}i}{56} ina he tāpiri te ±. Tāpiri -1 ki te i\sqrt{111}.

$$k=\frac{-1+\sqrt{111}i}{56}$$

Nā, me whakaoti te whārite k=\frac{-1±\sqrt{111}i}{56} ina he tango te ±. Tango i\sqrt{111} mai i -1.

$$k=\frac{-\sqrt{111}i-1}{56}$$

Kua oti te whārite te whakatau.

$$k=\frac{-1+\sqrt{111}i}{56} k=\frac{-\sqrt{111}i-1}{56}$$

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.

$$28k^{2}+k+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$28k^{2}+k+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$28k^{2}+k=-1$$

Whakawehea ngā taha e rua ki te 28.

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

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

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

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.

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

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

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

Tāpiri -\frac{1}{28} 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.

$$k^{2}+\frac{1}{28}k+\frac{1}{3136}=-\frac{111}{3136}$$

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

$$\left(k+\frac{1}{56}\right)^{2}=-\frac{111}{3136}$$

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

$$\sqrt{\left(k+\frac{1}{56}\right)^{2}}=\sqrt{-\frac{111}{3136}}$$

Whakarūnātia.

$$k+\frac{1}{56}=\frac{\sqrt{111}i}{56} k+\frac{1}{56}=-\frac{\sqrt{111}i}{56}$$

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

$$k=\frac{-1+\sqrt{111}i}{56} k=\frac{-\sqrt{111}i-1}{56}$$