Whakaoti i te 56x^2-12x+1=0

$$56 { x }^{ 2 } -12x+1=0$$

Whakaoti mō x (complex solution)

$$x=\frac{3+\sqrt{5}i}{28}\approx 0.107142857+0.079859571i$$

$$x=\frac{-\sqrt{5}i+3}{28}\approx 0.107142857-0.079859571i$$

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.

$$56x^{2}-12x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 56 mō a, -12 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 56}}{2\times 56}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 56}}{2\times 56}$$

Whakareatia -4 ki te 56.

$$x=\frac{-\left(-12\right)±\sqrt{144-224}}{2\times 56}$$

Tāpiri 144 ki te -224.

$$x=\frac{-\left(-12\right)±\sqrt{-80}}{2\times 56}$$

Tuhia te pūtakerua o te -80.

$$x=\frac{-\left(-12\right)±4\sqrt{5}i}{2\times 56}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±4\sqrt{5}i}{2\times 56}$$

Whakareatia 2 ki te 56.

$$x=\frac{12±4\sqrt{5}i}{112}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{5}i}{112} ina he tāpiri te ±. Tāpiri 12 ki te 4i\sqrt{5}.

$$x=\frac{12+4\sqrt{5}i}{112}$$

Whakawehe 12+4i\sqrt{5} ki te 112.

$$x=\frac{3+\sqrt{5}i}{28}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{5}i}{112} ina he tango te ±. Tango 4i\sqrt{5} mai i 12.

$$x=\frac{-4\sqrt{5}i+12}{112}$$

Whakawehe 12-4i\sqrt{5} ki te 112.

$$x=\frac{-\sqrt{5}i+3}{28}$$

Kua oti te whārite te whakatau.

$$x=\frac{3+\sqrt{5}i}{28} x=\frac{-\sqrt{5}i+3}{28}$$

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.

$$56x^{2}-12x+1=0$$

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

$$56x^{2}-12x+1-1=-1$$

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

$$56x^{2}-12x=-1$$

Whakawehea ngā taha e rua ki te 56.

$$\frac{56x^{2}-12x}{56}=-\frac{1}{56}$$

Mā te whakawehe ki te 56 ka wetekia te whakareanga ki te 56.

$$x^{2}+\left(-\frac{12}{56}\right)x=-\frac{1}{56}$$

Whakahekea te hautanga \frac{-12}{56} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{3}{14}x=-\frac{1}{56}$$

Whakawehea te -\frac{3}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{28}. Nā, tāpiria te pūrua o te -\frac{3}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{14}x+\left(-\frac{3}{28}\right)^{2}=-\frac{1}{56}+\left(-\frac{3}{28}\right)^{2}$$

Pūruatia -\frac{3}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{14}x+\frac{9}{784}=-\frac{1}{56}+\frac{9}{784}$$

Tāpiri -\frac{1}{56} ki te \frac{9}{784} 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.

$$x^{2}-\frac{3}{14}x+\frac{9}{784}=-\frac{5}{784}$$

Tauwehea x^{2}-\frac{3}{14}x+\frac{9}{784}. 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}.

$$\left(x-\frac{3}{28}\right)^{2}=-\frac{5}{784}$$

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

$$\sqrt{\left(x-\frac{3}{28}\right)^{2}}=\sqrt{-\frac{5}{784}}$$

Whakarūnātia.

$$x-\frac{3}{28}=\frac{\sqrt{5}i}{28} x-\frac{3}{28}=-\frac{\sqrt{5}i}{28}$$

Me tāpiri \frac{3}{28} ki ngā taha e rua o te whārite.

$$x=\frac{3+\sqrt{5}i}{28} x=\frac{-\sqrt{5}i+3}{28}$$