Whakaoti i te 80=6r+r^2

$$80 = 6 r + r ^ { 2 }$$

Whakaoti mō r (complex solution)

$$r=\sqrt{89}-3\approx 6.433981132$$

$$r=-\left(\sqrt{89}+3\right)\approx -12.433981132$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$6r+r^{2}=80$$

Tangohia te 80 mai i ngā taha e rua.

$$6r+r^{2}-80=0$$

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.

$$r^{2}+6r-80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me -80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-6±\sqrt{6^{2}-4\left(-80\right)}}{2}$$

Pūrua 6.

$$r=\frac{-6±\sqrt{36-4\left(-80\right)}}{2}$$

Whakareatia -4 ki te -80.

$$r=\frac{-6±\sqrt{36+320}}{2}$$

Tāpiri 36 ki te 320.

$$r=\frac{-6±\sqrt{356}}{2}$$

Tuhia te pūtakerua o te 356.

$$r=\frac{-6±2\sqrt{89}}{2}$$

Nā, me whakaoti te whārite r=\frac{-6±2\sqrt{89}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{89}.

$$r=\frac{2\sqrt{89}-6}{2}$$

Whakawehe -6+2\sqrt{89} ki te 2.

$$r=\sqrt{89}-3$$

Nā, me whakaoti te whārite r=\frac{-6±2\sqrt{89}}{2} ina he tango te ±. Tango 2\sqrt{89} mai i -6.

$$r=\frac{-2\sqrt{89}-6}{2}$$

Whakawehe -6-2\sqrt{89} ki te 2.

$$r=-\sqrt{89}-3$$

Kua oti te whārite te whakatau.

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$6r+r^{2}=80$$

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.

$$r^{2}+6r=80$$

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.

$$r^{2}+6r+3^{2}=80+3^{2}$$

Pūrua 3.

$$r^{2}+6r+9=80+9$$

Tāpiri 80 ki te 9.

$$r^{2}+6r+9=89$$

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

$$\left(r+3\right)^{2}=89$$

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

$$\sqrt{\left(r+3\right)^{2}}=\sqrt{89}$$

Whakarūnātia.

$$r+3=\sqrt{89} r+3=-\sqrt{89}$$

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

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$

Whakaoti mō r

$$r=\sqrt{89}-3\approx 6.433981132$$

$$r=-\sqrt{89}-3\approx -12.433981132$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$6r+r^{2}=80$$

Tangohia te 80 mai i ngā taha e rua.

$$6r+r^{2}-80=0$$

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.

$$r^{2}+6r-80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me -80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-6±\sqrt{6^{2}-4\left(-80\right)}}{2}$$

Pūrua 6.

$$r=\frac{-6±\sqrt{36-4\left(-80\right)}}{2}$$

Whakareatia -4 ki te -80.

$$r=\frac{-6±\sqrt{36+320}}{2}$$

Tāpiri 36 ki te 320.

$$r=\frac{-6±\sqrt{356}}{2}$$

Tuhia te pūtakerua o te 356.

$$r=\frac{-6±2\sqrt{89}}{2}$$

Nā, me whakaoti te whārite r=\frac{-6±2\sqrt{89}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{89}.

$$r=\frac{2\sqrt{89}-6}{2}$$

Whakawehe -6+2\sqrt{89} ki te 2.

$$r=\sqrt{89}-3$$

Nā, me whakaoti te whārite r=\frac{-6±2\sqrt{89}}{2} ina he tango te ±. Tango 2\sqrt{89} mai i -6.

$$r=\frac{-2\sqrt{89}-6}{2}$$

Whakawehe -6-2\sqrt{89} ki te 2.

$$r=-\sqrt{89}-3$$

Kua oti te whārite te whakatau.

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$6r+r^{2}=80$$

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.

$$r^{2}+6r=80$$

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.

$$r^{2}+6r+3^{2}=80+3^{2}$$

Pūrua 3.

$$r^{2}+6r+9=80+9$$

Tāpiri 80 ki te 9.

$$r^{2}+6r+9=89$$

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

$$\left(r+3\right)^{2}=89$$

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

$$\sqrt{\left(r+3\right)^{2}}=\sqrt{89}$$

Whakarūnātia.

$$r+3=\sqrt{89} r+3=-\sqrt{89}$$

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

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$