Whakaoti i te ex^2+3x+4=0

$$e x ^ { 2 } + 3 x + 4 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-3+i\sqrt{16e-9}}{2e}\approx -0.551819162+1.080283934i$$

$$x=-\frac{3+i\sqrt{16e-9}}{2e}\approx -0.551819162-1.080283934i$$

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.

$$ex^{2}+3x+4=0$$

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

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

Pūrua 3.

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

Whakareatia -4 ki te e.

$$x=\frac{-3±\sqrt{9+\left(-4e\right)\times 4}}{2e}$$

Whakareatia -4e ki te 4.

$$x=\frac{-3±\sqrt{9-16e}}{2e}$$

Tuhia te pūtakerua o te 9-16e.

$$x=\frac{-3±i\sqrt{-\left(9-16e\right)}}{2e}$$

Nā, me whakaoti te whārite x=\frac{-3±i\sqrt{-\left(9-16e\right)}}{2e} ina he tāpiri te ±. Tāpiri -3 ki te i\sqrt{-\left(9-16e\right)}.

$$x=\frac{-3+i\sqrt{16e-9}}{2e}$$

Nā, me whakaoti te whārite x=\frac{-3±i\sqrt{-\left(9-16e\right)}}{2e} ina he tango te ±. Tango i\sqrt{-\left(9-16e\right)} mai i -3.

$$x=\frac{-i\sqrt{16e-9}-3}{2e}$$

Whakawehe -3-i\sqrt{-9+16e} ki te 2e.

$$x=-\frac{3+i\sqrt{16e-9}}{2e}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3+i\sqrt{16e-9}}{2e} x=-\frac{3+i\sqrt{16e-9}}{2e}$$

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.

$$ex^{2}+3x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$ex^{2}+3x+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$ex^{2}+3x=-4$$

Whakawehea ngā taha e rua ki te e.

$$\frac{ex^{2}+3x}{e}=-\frac{4}{e}$$

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

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

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

$$x^{2}+\frac{3}{e}x+\left(\frac{3}{2e}\right)^{2}=-\frac{4}{e}+\left(\frac{3}{2e}\right)^{2}$$

Pūrua \frac{3}{2e}.

$$x^{2}+\frac{3}{e}x+\frac{9}{4e^{2}}=-\frac{4}{e}+\frac{9}{4e^{2}}$$

Tāpiri -\frac{4}{e} ki te \frac{9}{4e^{2}}.

$$x^{2}+\frac{3}{e}x+\frac{9}{4e^{2}}=\frac{\frac{9}{4}-4e}{e^{2}}$$

Tauwehea x^{2}+\frac{3}{e}x+\frac{9}{4e^{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}.

$$\left(x+\frac{3}{2e}\right)^{2}=\frac{\frac{9}{4}-4e}{e^{2}}$$

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

$$\sqrt{\left(x+\frac{3}{2e}\right)^{2}}=\sqrt{\frac{\frac{9}{4}-4e}{e^{2}}}$$

Whakarūnātia.

$$x+\frac{3}{2e}=\frac{i\sqrt{-\left(9-16e\right)}}{2e} x+\frac{3}{2e}=-\frac{i\sqrt{16e-9}}{2e}$$

Me tango \frac{3}{2e} mai i ngā taha e rua o te whārite.

$$x=\frac{-3+i\sqrt{16e-9}}{2e} x=-\frac{3+i\sqrt{16e-9}}{2e}$$