Whakaoti i te u^2-2/3u=5/4

$$u ^ { 2 } - \frac { 2 } { 3 } u = \frac { 5 } { 4 }$$

Whakaoti mō u

$$u=-\frac{5}{6}\approx -0.833333333$$

$$u = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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.

$$u^{2}-\frac{2}{3}u=\frac{5}{4}$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$u^{2}-\frac{2}{3}u-\frac{5}{4}=\frac{5}{4}-\frac{5}{4}$$

Mā te tango i te \frac{5}{4} i a ia ake anō ka toe ko te 0.

$$u^{2}-\frac{2}{3}u-\frac{5}{4}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -\frac{2}{3} mō b, me -\frac{5}{4} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\left(-\frac{2}{3}\right)^{2}-4\left(-\frac{5}{4}\right)}}{2}$$

Pūruatia -\frac{2}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{4}{9}-4\left(-\frac{5}{4}\right)}}{2}$$

Whakareatia -4 ki te -\frac{5}{4}.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{4}{9}+5}}{2}$$

Tāpiri \frac{4}{9} ki te 5.

$$u=\frac{-\left(-\frac{2}{3}\right)±\sqrt{\frac{49}{9}}}{2}$$

Tuhia te pūtakerua o te \frac{49}{9}.

$$u=\frac{-\left(-\frac{2}{3}\right)±\frac{7}{3}}{2}$$

Ko te tauaro o -\frac{2}{3} ko \frac{2}{3}.

$$u=\frac{\frac{2}{3}±\frac{7}{3}}{2}$$

Nā, me whakaoti te whārite u=\frac{\frac{2}{3}±\frac{7}{3}}{2} ina he tāpiri te ±. Tāpiri \frac{2}{3} ki te \frac{7}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$u=\frac{3}{2}$$

Nā, me whakaoti te whārite u=\frac{\frac{2}{3}±\frac{7}{3}}{2} ina he tango te ±. Tango \frac{7}{3} mai i \frac{2}{3} mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$u=-\frac{\frac{5}{3}}{2}$$

Whakawehe -\frac{5}{3} ki te 2.

$$u=-\frac{5}{6}$$

Kua oti te whārite te whakatau.

$$u=\frac{3}{2} u=-\frac{5}{6}$$

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.

$$u^{2}-\frac{2}{3}u=\frac{5}{4}$$

Whakawehea te -\frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{3}. Nā, tāpiria te pūrua o te -\frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$u^{2}-\frac{2}{3}u+\left(-\frac{1}{3}\right)^{2}=\frac{5}{4}+\left(-\frac{1}{3}\right)^{2}$$

Pūruatia -\frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$u^{2}-\frac{2}{3}u+\frac{1}{9}=\frac{5}{4}+\frac{1}{9}$$

Tāpiri \frac{5}{4} ki te \frac{1}{9} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$u^{2}-\frac{2}{3}u+\frac{1}{9}=\frac{49}{36}$$

Tauwehea te u^{2}-\frac{2}{3}u+\frac{1}{9}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(u-\frac{1}{3}\right)^{2}=\frac{49}{36}$$

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

$$\sqrt{\left(u-\frac{1}{3}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Whakarūnātia.

$$u-\frac{1}{3}=\frac{7}{6} u-\frac{1}{3}=-\frac{7}{6}$$

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

$$u=\frac{3}{2} u=-\frac{5}{6}$$