Whakaoti i te 7x^2+4x+1=0

$$7 x ^ { 2 } + 4 x + 1 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-2+\sqrt{3}i}{7}\approx -0.285714286+0.24743583i$$

$$x=\frac{-\sqrt{3}i-2}{7}\approx -0.285714286-0.24743583i$$

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.

$$7x^{2}+4x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 4 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 7}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-4±\sqrt{16-28}}{2\times 7}$$

Tāpiri 16 ki te -28.

$$x=\frac{-4±\sqrt{-12}}{2\times 7}$$

Tuhia te pūtakerua o te -12.

$$x=\frac{-4±2\sqrt{3}i}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-4±2\sqrt{3}i}{14}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{3}i}{14} ina he tāpiri te ±. Tāpiri -4 ki te 2i\sqrt{3}.

$$x=\frac{-4+2\sqrt{3}i}{14}$$

Whakawehe -4+2i\sqrt{3} ki te 14.

$$x=\frac{-2+\sqrt{3}i}{7}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{3}i}{14} ina he tango te ±. Tango 2i\sqrt{3} mai i -4.

$$x=\frac{-2\sqrt{3}i-4}{14}$$

Whakawehe -4-2i\sqrt{3} ki te 14.

$$x=\frac{-\sqrt{3}i-2}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{-2+\sqrt{3}i}{7} x=\frac{-\sqrt{3}i-2}{7}$$

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.

$$7x^{2}+4x+1=0$$

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

$$7x^{2}+4x+1-1=-1$$

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

$$7x^{2}+4x=-1$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}+4x}{7}=-\frac{1}{7}$$

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

$$x^{2}+\frac{4}{7}x=-\frac{1}{7}$$

Whakawehea te \frac{4}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{2}{7}. Nā, tāpiria te pūrua o te \frac{2}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{4}{7}x+\left(\frac{2}{7}\right)^{2}=-\frac{1}{7}+\left(\frac{2}{7}\right)^{2}$$

Pūruatia \frac{2}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=-\frac{1}{7}+\frac{4}{49}$$

Tāpiri -\frac{1}{7} ki te \frac{4}{49} 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.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=-\frac{3}{49}$$

Tauwehea te x^{2}+\frac{4}{7}x+\frac{4}{49}. 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}.

$$\left(x+\frac{2}{7}\right)^{2}=-\frac{3}{49}$$

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

$$\sqrt{\left(x+\frac{2}{7}\right)^{2}}=\sqrt{-\frac{3}{49}}$$

Whakarūnātia.

$$x+\frac{2}{7}=\frac{\sqrt{3}i}{7} x+\frac{2}{7}=-\frac{\sqrt{3}i}{7}$$

Me tango \frac{2}{7} mai i ngā taha e rua o te whārite.

$$x=\frac{-2+\sqrt{3}i}{7} x=\frac{-\sqrt{3}i-2}{7}$$