Whakaoti i te 5r^2-44r+120=-30

$$5 r ^ { 2 } - 44 r + 120 = - 30$$

Whakaoti mō r

$$r=\frac{22+\sqrt{266}i}{5}\approx 4.4+3.261901286i$$

$$r=\frac{-\sqrt{266}i+22}{5}\approx 4.4-3.261901286i$$

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.

$$5r^{2}-44r+120=-30$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$5r^{2}-44r+120-\left(-30\right)=-30-\left(-30\right)$$

Mā te tango i te -30 i a ia ake anō ka toe ko te 0.

$$5r^{2}-44r+120-\left(-30\right)=0$$

Tango -30 mai i 120.

$$5r^{2}-44r+150=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -44 mō b, me 150 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 5\times 150}}{2\times 5}$$

Pūrua -44.

$$r=\frac{-\left(-44\right)±\sqrt{1936-4\times 5\times 150}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$r=\frac{-\left(-44\right)±\sqrt{1936-20\times 150}}{2\times 5}$$

Whakareatia -20 ki te 150.

$$r=\frac{-\left(-44\right)±\sqrt{1936-3000}}{2\times 5}$$

Tāpiri 1936 ki te -3000.

$$r=\frac{-\left(-44\right)±\sqrt{-1064}}{2\times 5}$$

Tuhia te pūtakerua o te -1064.

$$r=\frac{-\left(-44\right)±2\sqrt{266}i}{2\times 5}$$

Ko te tauaro o -44 ko 44.

$$r=\frac{44±2\sqrt{266}i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$r=\frac{44±2\sqrt{266}i}{10}$$

Nā, me whakaoti te whārite r=\frac{44±2\sqrt{266}i}{10} ina he tāpiri te ±. Tāpiri 44 ki te 2i\sqrt{266}.

$$r=\frac{44+2\sqrt{266}i}{10}$$

Whakawehe 44+2i\sqrt{266} ki te 10.

$$r=\frac{22+\sqrt{266}i}{5}$$

Nā, me whakaoti te whārite r=\frac{44±2\sqrt{266}i}{10} ina he tango te ±. Tango 2i\sqrt{266} mai i 44.

$$r=\frac{-2\sqrt{266}i+44}{10}$$

Whakawehe 44-2i\sqrt{266} ki te 10.

$$r=\frac{-\sqrt{266}i+22}{5}$$

Kua oti te whārite te whakatau.

$$r=\frac{22+\sqrt{266}i}{5} r=\frac{-\sqrt{266}i+22}{5}$$

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.

$$5r^{2}-44r+120=-30$$

Me tango 120 mai i ngā taha e rua o te whārite.

$$5r^{2}-44r+120-120=-30-120$$

Mā te tango i te 120 i a ia ake anō ka toe ko te 0.

$$5r^{2}-44r=-30-120$$

Tango 120 mai i -30.

$$5r^{2}-44r=-150$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5r^{2}-44r}{5}=-\frac{150}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$r^{2}-\frac{44}{5}r=-\frac{150}{5}$$

Whakawehe -150 ki te 5.

$$r^{2}-\frac{44}{5}r=-30$$

Whakawehea te -\frac{44}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{22}{5}. Nā, tāpiria te pūrua o te -\frac{22}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}-\frac{44}{5}r+\left(-\frac{22}{5}\right)^{2}=-30+\left(-\frac{22}{5}\right)^{2}$$

Pūruatia -\frac{22}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$r^{2}-\frac{44}{5}r+\frac{484}{25}=-30+\frac{484}{25}$$

Tāpiri -30 ki te \frac{484}{25}.

$$r^{2}-\frac{44}{5}r+\frac{484}{25}=-\frac{266}{25}$$

Tauwehea r^{2}-\frac{44}{5}r+\frac{484}{25}. 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}.

$$\left(r-\frac{22}{5}\right)^{2}=-\frac{266}{25}$$

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

$$\sqrt{\left(r-\frac{22}{5}\right)^{2}}=\sqrt{-\frac{266}{25}}$$

Whakarūnātia.

$$r-\frac{22}{5}=\frac{\sqrt{266}i}{5} r-\frac{22}{5}=-\frac{\sqrt{266}i}{5}$$

Me tāpiri \frac{22}{5} ki ngā taha e rua o te whārite.

$$r=\frac{22+\sqrt{266}i}{5} r=\frac{-\sqrt{266}i+22}{5}$$