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

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

Whakaoti mō y (complex solution)

$$y=\sqrt{7}-1\approx 1.645751311$$

$$y=-\left(\sqrt{7}+1\right)\approx -3.645751311$$

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.

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

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

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

Pūrua 2.

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

Whakareatia -4 ki te -6.

$$y=\frac{-2±\sqrt{4+24}}{2}$$

Tāpiri 4 ki te 24.

$$y=\frac{-2±\sqrt{28}}{2}$$

Tuhia te pūtakerua o te 28.

$$y=\frac{-2±2\sqrt{7}}{2}$$

Nā, me whakaoti te whārite y=\frac{-2±2\sqrt{7}}{2} ina he tāpiri te ±. Tāpiri -2 ki te 2\sqrt{7}.

$$y=\frac{2\sqrt{7}-2}{2}$$

Whakawehe -2+2\sqrt{7} ki te 2.

$$y=\sqrt{7}-1$$

Nā, me whakaoti te whārite y=\frac{-2±2\sqrt{7}}{2} ina he tango te ±. Tango 2\sqrt{7} mai i -2.

$$y=\frac{-2\sqrt{7}-2}{2}$$

Whakawehe -2-2\sqrt{7} ki te 2.

$$y=-\sqrt{7}-1$$

Kua oti te whārite te whakatau.

$$y=\sqrt{7}-1 y=-\sqrt{7}-1$$

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.

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

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

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

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

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

Tango -6 mai i 0.

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

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+2y+1^{2}=6+1^{2}$$

Pūrua 1.

$$y^{2}+2y+1=6+1$$

Tāpiri 6 ki te 1.

$$y^{2}+2y+1=7$$

Tauwehea y^{2}+2y+1. 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}.

$$\left(y+1\right)^{2}=7$$

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

$$\sqrt{\left(y+1\right)^{2}}=\sqrt{7}$$

Whakarūnātia.

$$y+1=\sqrt{7} y+1=-\sqrt{7}$$

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

$$y=\sqrt{7}-1 y=-\sqrt{7}-1$$

Whakaoti mō y

$$y=\sqrt{7}-1\approx 1.645751311$$

$$y=-\sqrt{7}-1\approx -3.645751311$$

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.

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

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

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

Pūrua 2.

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

Whakareatia -4 ki te -6.

$$y=\frac{-2±\sqrt{4+24}}{2}$$

Tāpiri 4 ki te 24.

$$y=\frac{-2±\sqrt{28}}{2}$$

Tuhia te pūtakerua o te 28.

$$y=\frac{-2±2\sqrt{7}}{2}$$

Nā, me whakaoti te whārite y=\frac{-2±2\sqrt{7}}{2} ina he tāpiri te ±. Tāpiri -2 ki te 2\sqrt{7}.

$$y=\frac{2\sqrt{7}-2}{2}$$

Whakawehe -2+2\sqrt{7} ki te 2.

$$y=\sqrt{7}-1$$

Nā, me whakaoti te whārite y=\frac{-2±2\sqrt{7}}{2} ina he tango te ±. Tango 2\sqrt{7} mai i -2.

$$y=\frac{-2\sqrt{7}-2}{2}$$

Whakawehe -2-2\sqrt{7} ki te 2.

$$y=-\sqrt{7}-1$$

Kua oti te whārite te whakatau.

$$y=\sqrt{7}-1 y=-\sqrt{7}-1$$

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.

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

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

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

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

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

Tango -6 mai i 0.

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

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+2y+1^{2}=6+1^{2}$$

Pūrua 1.

$$y^{2}+2y+1=6+1$$

Tāpiri 6 ki te 1.

$$y^{2}+2y+1=7$$

Tauwehea y^{2}+2y+1. 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}.

$$\left(y+1\right)^{2}=7$$

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

$$\sqrt{\left(y+1\right)^{2}}=\sqrt{7}$$

Whakarūnātia.

$$y+1=\sqrt{7} y+1=-\sqrt{7}$$

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

$$y=\sqrt{7}-1 y=-\sqrt{7}-1$$