Whakaoti i te 0=y^2+6y-14

$$0 = y ^ { 2 } + 6 y - 14$$

Whakaoti mō y (complex solution)

$$y=\sqrt{23}-3\approx 1.795831523$$

$$y=-\left(\sqrt{23}+3\right)\approx -7.795831523$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$y^{2}+6y-14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me -14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-14\right)}}{2}$$

Pūrua 6.

$$y=\frac{-6±\sqrt{36-4\left(-14\right)}}{2}$$

Whakareatia -4 ki te -14.

$$y=\frac{-6±\sqrt{36+56}}{2}$$

Tāpiri 36 ki te 56.

$$y=\frac{-6±\sqrt{92}}{2}$$

Tuhia te pūtakerua o te 92.

$$y=\frac{-6±2\sqrt{23}}{2}$$

Nā, me whakaoti te whārite y=\frac{-6±2\sqrt{23}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{23}.

$$y=\frac{2\sqrt{23}-6}{2}$$

Whakawehe -6+2\sqrt{23} ki te 2.

$$y=\sqrt{23}-3$$

Nā, me whakaoti te whārite y=\frac{-6±2\sqrt{23}}{2} ina he tango te ±. Tango 2\sqrt{23} mai i -6.

$$y=\frac{-2\sqrt{23}-6}{2}$$

Whakawehe -6-2\sqrt{23} ki te 2.

$$y=-\sqrt{23}-3$$

Kua oti te whārite te whakatau.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$y^{2}+6y-14=0$$

Me tāpiri te 14 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y^{2}+6y=14$$

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.

$$y^{2}+6y+3^{2}=14+3^{2}$$

Pūrua 3.

$$y^{2}+6y+9=14+9$$

Tāpiri 14 ki te 9.

$$y^{2}+6y+9=23$$

Tauwehea y^{2}+6y+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}.

$$\left(y+3\right)^{2}=23$$

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

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{23}$$

Whakarūnātia.

$$y+3=\sqrt{23} y+3=-\sqrt{23}$$

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

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Whakaoti mō y

$$y=\sqrt{23}-3\approx 1.795831523$$

$$y=-\sqrt{23}-3\approx -7.795831523$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$y^{2}+6y-14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me -14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-14\right)}}{2}$$

Pūrua 6.

$$y=\frac{-6±\sqrt{36-4\left(-14\right)}}{2}$$

Whakareatia -4 ki te -14.

$$y=\frac{-6±\sqrt{36+56}}{2}$$

Tāpiri 36 ki te 56.

$$y=\frac{-6±\sqrt{92}}{2}$$

Tuhia te pūtakerua o te 92.

$$y=\frac{-6±2\sqrt{23}}{2}$$

Nā, me whakaoti te whārite y=\frac{-6±2\sqrt{23}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{23}.

$$y=\frac{2\sqrt{23}-6}{2}$$

Whakawehe -6+2\sqrt{23} ki te 2.

$$y=\sqrt{23}-3$$

Nā, me whakaoti te whārite y=\frac{-6±2\sqrt{23}}{2} ina he tango te ±. Tango 2\sqrt{23} mai i -6.

$$y=\frac{-2\sqrt{23}-6}{2}$$

Whakawehe -6-2\sqrt{23} ki te 2.

$$y=-\sqrt{23}-3$$

Kua oti te whārite te whakatau.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$y^{2}+6y-14=0$$

Me tāpiri te 14 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y^{2}+6y=14$$

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.

$$y^{2}+6y+3^{2}=14+3^{2}$$

Pūrua 3.

$$y^{2}+6y+9=14+9$$

Tāpiri 14 ki te 9.

$$y^{2}+6y+9=23$$

Tauwehea y^{2}+6y+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}.

$$\left(y+3\right)^{2}=23$$

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

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{23}$$

Whakarūnātia.

$$y+3=\sqrt{23} y+3=-\sqrt{23}$$

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

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$