Whakaoti i te x^2+6x+3=0

$${ x }^{ 2 } +6x+3=0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{6}-3\approx -0.550510257$$

$$x=-\left(\sqrt{6}+3\right)\approx -5.449489743$$

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.

$$x^{2}+6x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3}}{2}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 3}}{2}$$

Whakareatia -4 ki te 3.

$$x=\frac{-6±\sqrt{36-12}}{2}$$

Tāpiri 36 ki te -12.

$$x=\frac{-6±\sqrt{24}}{2}$$

Tuhia te pūtakerua o te 24.

$$x=\frac{-6±2\sqrt{6}}{2}$$

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

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

Whakawehe -6+2\sqrt{6} ki te 2.

$$x=\sqrt{6}-3$$

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

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

Whakawehe -6-2\sqrt{6} ki te 2.

$$x=-\sqrt{6}-3$$

Kua oti te whārite te whakatau.

$$x=\sqrt{6}-3 x=-\sqrt{6}-3$$

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.

$$x^{2}+6x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x^{2}+6x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$x^{2}+6x=-3$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+6x+3^{2}=-3+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=-3+9$$

Tāpiri -3 ki te 9.

$$x^{2}+6x+9=6$$

Tauwehea x^{2}+6x+9. 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}.

$$\left(x+3\right)^{2}=6$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{6}$$

Whakarūnātia.

$$x+3=\sqrt{6} x+3=-\sqrt{6}$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=\sqrt{6}-3 x=-\sqrt{6}-3$$

Whakaoti mō x

$$x=\sqrt{6}-3\approx -0.550510257$$

$$x=-\sqrt{6}-3\approx -5.449489743$$

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.

$$x^{2}+6x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3}}{2}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 3}}{2}$$

Whakareatia -4 ki te 3.

$$x=\frac{-6±\sqrt{36-12}}{2}$$

Tāpiri 36 ki te -12.

$$x=\frac{-6±\sqrt{24}}{2}$$

Tuhia te pūtakerua o te 24.

$$x=\frac{-6±2\sqrt{6}}{2}$$

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

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

Whakawehe -6+2\sqrt{6} ki te 2.

$$x=\sqrt{6}-3$$

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

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

Whakawehe -6-2\sqrt{6} ki te 2.

$$x=-\sqrt{6}-3$$

Kua oti te whārite te whakatau.

$$x=\sqrt{6}-3 x=-\sqrt{6}-3$$

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.

$$x^{2}+6x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x^{2}+6x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$x^{2}+6x=-3$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+6x+3^{2}=-3+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=-3+9$$

Tāpiri -3 ki te 9.

$$x^{2}+6x+9=6$$

Tauwehea x^{2}+6x+9. 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}.

$$\left(x+3\right)^{2}=6$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{6}$$

Whakarūnātia.

$$x+3=\sqrt{6} x+3=-\sqrt{6}$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=\sqrt{6}-3 x=-\sqrt{6}-3$$