Whakaoti i te x^2+sqrt{6}x+5=0

$$x ^ { 2 } + \sqrt { 6 } x + 5 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{6}+\sqrt{14}i}{2}\approx -1.224744871+1.870828693i$$

$$x=\frac{-\sqrt{14}i-\sqrt{6}}{2}\approx -1.224744871-1.870828693i$$

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.

$$x^{2}+\sqrt{6}x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, \sqrt{6} mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\sqrt{6}±\sqrt{\left(\sqrt{6}\right)^{2}-4\times 5}}{2}$$

Pūrua \sqrt{6}.

$$x=\frac{-\sqrt{6}±\sqrt{6-4\times 5}}{2}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\sqrt{6}±\sqrt{6-20}}{2}$$

Tāpiri 6 ki te -20.

$$x=\frac{-\sqrt{6}±\sqrt{-14}}{2}$$

Tuhia te pūtakerua o te -14.

$$x=\frac{-\sqrt{6}±\sqrt{14}i}{2}$$

Nā, me whakaoti te whārite x=\frac{-\sqrt{6}±\sqrt{14}i}{2} ina he tāpiri te ±. Tāpiri -\sqrt{6} ki te i\sqrt{14}.

$$x=\frac{-\sqrt{6}+\sqrt{14}i}{2}$$

Nā, me whakaoti te whārite x=\frac{-\sqrt{6}±\sqrt{14}i}{2} ina he tango te ±. Tango i\sqrt{14} mai i -\sqrt{6}.

$$x=\frac{-\sqrt{14}i-\sqrt{6}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{6}+\sqrt{14}i}{2} x=\frac{-\sqrt{14}i-\sqrt{6}}{2}$$

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.

$$x^{2}+\sqrt{6}x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x^{2}+\sqrt{6}x+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$x^{2}+\sqrt{6}x=-5$$

Whakawehea te \sqrt{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{\sqrt{6}}{2}. Nā, tāpiria te pūrua o te \frac{\sqrt{6}}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\sqrt{6}x+\left(\frac{\sqrt{6}}{2}\right)^{2}=-5+\left(\frac{\sqrt{6}}{2}\right)^{2}$$

Pūrua \frac{\sqrt{6}}{2}.

$$x^{2}+\sqrt{6}x+\frac{3}{2}=-5+\frac{3}{2}$$

Tāpiri -5 ki te \frac{3}{2}.

$$x^{2}+\sqrt{6}x+\frac{3}{2}=-\frac{7}{2}$$

Tauwehea x^{2}+\sqrt{6}x+\frac{3}{2}. 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}.

$$\left(x+\frac{\sqrt{6}}{2}\right)^{2}=-\frac{7}{2}$$

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

$$\sqrt{\left(x+\frac{\sqrt{6}}{2}\right)^{2}}=\sqrt{-\frac{7}{2}}$$

Whakarūnātia.

$$x+\frac{\sqrt{6}}{2}=\frac{\sqrt{14}i}{2} x+\frac{\sqrt{6}}{2}=-\frac{\sqrt{14}i}{2}$$

Me tango \frac{\sqrt{6}}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{-\sqrt{6}+\sqrt{14}i}{2} x=\frac{-\sqrt{14}i-\sqrt{6}}{2}$$