Whakaoti i te 5x^2+3x+1=0

$$5 x ^ { 2 } + 3 x + 1 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-3+\sqrt{11}i}{10}\approx -0.3+0.331662479i$$

$$x=\frac{-\sqrt{11}i-3}{10}\approx -0.3-0.331662479i$$

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.

$$5x^{2}+3x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 3 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 5}}{2\times 5}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\times 5}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-3±\sqrt{9-20}}{2\times 5}$$

Tāpiri 9 ki te -20.

$$x=\frac{-3±\sqrt{-11}}{2\times 5}$$

Tuhia te pūtakerua o te -11.

$$x=\frac{-3±\sqrt{11}i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-3±\sqrt{11}i}{10}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{11}i}{10} ina he tāpiri te ±. Tāpiri -3 ki te i\sqrt{11}.

$$x=\frac{-3+\sqrt{11}i}{10}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{11}i}{10} ina he tango te ±. Tango i\sqrt{11} mai i -3.

$$x=\frac{-\sqrt{11}i-3}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i-3}{10}$$

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.

$$5x^{2}+3x+1=0$$

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

$$5x^{2}+3x+1-1=-1$$

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

$$5x^{2}+3x=-1$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+3x}{5}=-\frac{1}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\frac{3}{5}x=-\frac{1}{5}$$

Whakawehea te \frac{3}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{10}. Nā, tāpiria te pūrua o te \frac{3}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{5}x+\left(\frac{3}{10}\right)^{2}=-\frac{1}{5}+\left(\frac{3}{10}\right)^{2}$$

Pūruatia \frac{3}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-\frac{1}{5}+\frac{9}{100}$$

Tāpiri -\frac{1}{5} ki te \frac{9}{100} 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.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-\frac{11}{100}$$

Tauwehea te x^{2}+\frac{3}{5}x+\frac{9}{100}. 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}.

$$\left(x+\frac{3}{10}\right)^{2}=-\frac{11}{100}$$

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

$$\sqrt{\left(x+\frac{3}{10}\right)^{2}}=\sqrt{-\frac{11}{100}}$$

Whakarūnātia.

$$x+\frac{3}{10}=\frac{\sqrt{11}i}{10} x+\frac{3}{10}=-\frac{\sqrt{11}i}{10}$$

Me tango \frac{3}{10} mai i ngā taha e rua o te whārite.

$$x=\frac{-3+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i-3}{10}$$