Whakaoti i te 5m^2-14m-15=0

$$5 m ^ { 2 } - 14 m - 15 = 0$$

Whakaoti mō m

$$m = \frac{2 \sqrt{31} + 7}{5} \approx 3.627105745$$

$$m=\frac{7-2\sqrt{31}}{5}\approx -0.827105745$$

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.

$$5m^{2}-14m-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -14 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\left(-15\right)}}{2\times 5}$$

Pūrua -14.

$$m=\frac{-\left(-14\right)±\sqrt{196-4\times 5\left(-15\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$m=\frac{-\left(-14\right)±\sqrt{196-20\left(-15\right)}}{2\times 5}$$

Whakareatia -20 ki te -15.

$$m=\frac{-\left(-14\right)±\sqrt{196+300}}{2\times 5}$$

Tāpiri 196 ki te 300.

$$m=\frac{-\left(-14\right)±\sqrt{496}}{2\times 5}$$

Tuhia te pūtakerua o te 496.

$$m=\frac{-\left(-14\right)±4\sqrt{31}}{2\times 5}$$

Ko te tauaro o -14 ko 14.

$$m=\frac{14±4\sqrt{31}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$m=\frac{14±4\sqrt{31}}{10}$$

Nā, me whakaoti te whārite m=\frac{14±4\sqrt{31}}{10} ina he tāpiri te ±. Tāpiri 14 ki te 4\sqrt{31}.

$$m=\frac{4\sqrt{31}+14}{10}$$

Whakawehe 14+4\sqrt{31} ki te 10.

$$m=\frac{2\sqrt{31}+7}{5}$$

Nā, me whakaoti te whārite m=\frac{14±4\sqrt{31}}{10} ina he tango te ±. Tango 4\sqrt{31} mai i 14.

$$m=\frac{14-4\sqrt{31}}{10}$$

Whakawehe 14-4\sqrt{31} ki te 10.

$$m=\frac{7-2\sqrt{31}}{5}$$

Kua oti te whārite te whakatau.

$$m=\frac{2\sqrt{31}+7}{5} m=\frac{7-2\sqrt{31}}{5}$$

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.

$$5m^{2}-14m-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$5m^{2}-14m-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$5m^{2}-14m=-\left(-15\right)$$

Tango -15 mai i 0.

$$5m^{2}-14m=15$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5m^{2}-14m}{5}=\frac{15}{5}$$

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

$$m^{2}-\frac{14}{5}m=\frac{15}{5}$$

Whakawehe 15 ki te 5.

$$m^{2}-\frac{14}{5}m=3$$

Whakawehea te -\frac{14}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{5}. Nā, tāpiria te pūrua o te -\frac{7}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$m^{2}-\frac{14}{5}m+\left(-\frac{7}{5}\right)^{2}=3+\left(-\frac{7}{5}\right)^{2}$$

Pūruatia -\frac{7}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$m^{2}-\frac{14}{5}m+\frac{49}{25}=3+\frac{49}{25}$$

Tāpiri 3 ki te \frac{49}{25}.

$$m^{2}-\frac{14}{5}m+\frac{49}{25}=\frac{124}{25}$$

Tauwehea m^{2}-\frac{14}{5}m+\frac{49}{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}.

$$\left(m-\frac{7}{5}\right)^{2}=\frac{124}{25}$$

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

$$\sqrt{\left(m-\frac{7}{5}\right)^{2}}=\sqrt{\frac{124}{25}}$$

Whakarūnātia.

$$m-\frac{7}{5}=\frac{2\sqrt{31}}{5} m-\frac{7}{5}=-\frac{2\sqrt{31}}{5}$$

Me tāpiri \frac{7}{5} ki ngā taha e rua o te whārite.

$$m=\frac{2\sqrt{31}+7}{5} m=\frac{7-2\sqrt{31}}{5}$$