Whakaoti i te 1/15x^2-3/10x+1/3=0

$$\frac { 1 } { 15 } x ^ { 2 } - \frac { 3 } { 10 } x + \frac { 1 } { 3 } = 0$$

Whakaoti mō x

$$x=2$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

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.

$$\frac{1}{15}x^{2}-\frac{3}{10}x+\frac{1}{3}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{15} mō a, -\frac{3}{10} mō b, me \frac{1}{3} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\left(-\frac{3}{10}\right)^{2}-4\times \frac{1}{15}\times \frac{1}{3}}}{2\times \frac{1}{15}}$$

Pūruatia -\frac{3}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-4\times \frac{1}{15}\times \frac{1}{3}}}{2\times \frac{1}{15}}$$

Whakareatia -4 ki te \frac{1}{15}.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-\frac{4}{15}\times \frac{1}{3}}}{2\times \frac{1}{15}}$$

Whakareatia -\frac{4}{15} ki te \frac{1}{3} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-\frac{4}{45}}}{2\times \frac{1}{15}}$$

Tāpiri \frac{9}{100} ki te -\frac{4}{45} 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.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{1}{900}}}{2\times \frac{1}{15}}$$

Tuhia te pūtakerua o te \frac{1}{900}.

$$x=\frac{-\left(-\frac{3}{10}\right)±\frac{1}{30}}{2\times \frac{1}{15}}$$

Ko te tauaro o -\frac{3}{10} ko \frac{3}{10}.

$$x=\frac{\frac{3}{10}±\frac{1}{30}}{2\times \frac{1}{15}}$$

Whakareatia 2 ki te \frac{1}{15}.

$$x=\frac{\frac{3}{10}±\frac{1}{30}}{\frac{2}{15}}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{10}±\frac{1}{30}}{\frac{2}{15}} ina he tāpiri te ±. Tāpiri \frac{3}{10} ki te \frac{1}{30} 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.

$$x=\frac{\frac{1}{3}}{\frac{2}{15}}$$

Whakawehe \frac{1}{3} ki te \frac{2}{15} mā te whakarea \frac{1}{3} ki te tau huripoki o \frac{2}{15}.

$$x=\frac{5}{2}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{10}±\frac{1}{30}}{\frac{2}{15}} ina he tango te ±. Tango \frac{1}{30} mai i \frac{3}{10} 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.

$$x=\frac{\frac{4}{15}}{\frac{2}{15}}$$

Whakawehe \frac{4}{15} ki te \frac{2}{15} mā te whakarea \frac{4}{15} ki te tau huripoki o \frac{2}{15}.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{2} x=2$$

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.

$$\frac{1}{15}x^{2}-\frac{3}{10}x+\frac{1}{3}=0$$

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$\frac{1}{15}x^{2}-\frac{3}{10}x+\frac{1}{3}-\frac{1}{3}=-\frac{1}{3}$$

Mā te tango i te \frac{1}{3} i a ia ake anō ka toe ko te 0.

$$\frac{1}{15}x^{2}-\frac{3}{10}x=-\frac{1}{3}$$

Me whakarea ngā taha e rua ki te 15.

$$\frac{\frac{1}{15}x^{2}-\frac{3}{10}x}{\frac{1}{15}}=-\frac{\frac{1}{3}}{\frac{1}{15}}$$

Mā te whakawehe ki te \frac{1}{15} ka wetekia te whakareanga ki te \frac{1}{15}.

$$x^{2}+\left(-\frac{\frac{3}{10}}{\frac{1}{15}}\right)x=-\frac{\frac{1}{3}}{\frac{1}{15}}$$

Whakawehe -\frac{3}{10} ki te \frac{1}{15} mā te whakarea -\frac{3}{10} ki te tau huripoki o \frac{1}{15}.

$$x^{2}-\frac{9}{2}x=-\frac{\frac{1}{3}}{\frac{1}{15}}$$

Whakawehe -\frac{1}{3} ki te \frac{1}{15} mā te whakarea -\frac{1}{3} ki te tau huripoki o \frac{1}{15}.

$$x^{2}-\frac{9}{2}x=-5$$

Whakawehea te -\frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{4}. Nā, tāpiria te pūrua o te -\frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-5+\left(-\frac{9}{4}\right)^{2}$$

Pūruatia -\frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-5+\frac{81}{16}$$

Tāpiri -5 ki te \frac{81}{16}.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{1}{16}$$

Tauwehea te x^{2}-\frac{9}{2}x+\frac{81}{16}. 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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{1}{16}$$

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

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Whakarūnātia.

$$x-\frac{9}{4}=\frac{1}{4} x-\frac{9}{4}=-\frac{1}{4}$$

Me tāpiri \frac{9}{4} ki ngā taha e rua o te whārite.

$$x=\frac{5}{2} x=2$$