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

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

Whakaoti mō x (complex solution)

$$x=\sqrt{15}-3\approx 0.872983346$$

$$x=-\left(\sqrt{15}+3\right)\approx -6.872983346$$

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.

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

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

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

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-6\right)}}{2}$$

Whakareatia -4 ki te -6.

$$x=\frac{-6±\sqrt{36+24}}{2}$$

Tāpiri 36 ki te 24.

$$x=\frac{-6±\sqrt{60}}{2}$$

Tuhia te pūtakerua o te 60.

$$x=\frac{-6±2\sqrt{15}}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{15}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-6}{2}$$

Whakawehe -6+2\sqrt{15} ki te 2.

$$x=\sqrt{15}-3$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{15}}{2} ina he tango te ±. Tango 2\sqrt{15} mai i -6.

$$x=\frac{-2\sqrt{15}-6}{2}$$

Whakawehe -6-2\sqrt{15} ki te 2.

$$x=-\sqrt{15}-3$$

Kua oti te whārite te whakatau.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$

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.

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

Me tāpiri 6 ki ngā taha e rua o te whārite.

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

Mā te tango i te -6 i a ia ake anō ka toe ko te 0.

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

Tango -6 mai i 0.

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

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.

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

Pūrua 3.

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

Tāpiri 6 ki te 9.

$$x^{2}+6x+9=15$$

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}.

$$\left(x+3\right)^{2}=15$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{15}$$

Whakarūnātia.

$$x+3=\sqrt{15} x+3=-\sqrt{15}$$

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

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$

Whakaoti mō x

$$x=\sqrt{15}-3\approx 0.872983346$$

$$x=-\sqrt{15}-3\approx -6.872983346$$

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.

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

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

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

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-6\right)}}{2}$$

Whakareatia -4 ki te -6.

$$x=\frac{-6±\sqrt{36+24}}{2}$$

Tāpiri 36 ki te 24.

$$x=\frac{-6±\sqrt{60}}{2}$$

Tuhia te pūtakerua o te 60.

$$x=\frac{-6±2\sqrt{15}}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{15}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-6}{2}$$

Whakawehe -6+2\sqrt{15} ki te 2.

$$x=\sqrt{15}-3$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{15}}{2} ina he tango te ±. Tango 2\sqrt{15} mai i -6.

$$x=\frac{-2\sqrt{15}-6}{2}$$

Whakawehe -6-2\sqrt{15} ki te 2.

$$x=-\sqrt{15}-3$$

Kua oti te whārite te whakatau.

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$

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.

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

Me tāpiri 6 ki ngā taha e rua o te whārite.

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

Mā te tango i te -6 i a ia ake anō ka toe ko te 0.

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

Tango -6 mai i 0.

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

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.

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

Pūrua 3.

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

Tāpiri 6 ki te 9.

$$x^{2}+6x+9=15$$

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}.

$$\left(x+3\right)^{2}=15$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{15}$$

Whakarūnātia.

$$x+3=\sqrt{15} x+3=-\sqrt{15}$$

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

$$x=\sqrt{15}-3 x=-\sqrt{15}-3$$