Whakaoti i te 81b^2-126b+48=0

$$81 b ^ { 2 } - 126 b + 48 = 0$$

Whakaoti mō b

$$b=\frac{2}{3}\approx 0.666666667$$

$$b=\frac{8}{9}\approx 0.888888889$$

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.

$$81b^{2}-126b+48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 81 mō a, -126 mō b, me 48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-126\right)±\sqrt{\left(-126\right)^{2}-4\times 81\times 48}}{2\times 81}$$

Pūrua -126.

$$b=\frac{-\left(-126\right)±\sqrt{15876-4\times 81\times 48}}{2\times 81}$$

Whakareatia -4 ki te 81.

$$b=\frac{-\left(-126\right)±\sqrt{15876-324\times 48}}{2\times 81}$$

Whakareatia -324 ki te 48.

$$b=\frac{-\left(-126\right)±\sqrt{15876-15552}}{2\times 81}$$

Tāpiri 15876 ki te -15552.

$$b=\frac{-\left(-126\right)±\sqrt{324}}{2\times 81}$$

Tuhia te pūtakerua o te 324.

$$b=\frac{-\left(-126\right)±18}{2\times 81}$$

Ko te tauaro o -126 ko 126.

$$b=\frac{126±18}{2\times 81}$$

Whakareatia 2 ki te 81.

$$b=\frac{126±18}{162}$$

Nā, me whakaoti te whārite b=\frac{126±18}{162} ina he tāpiri te ±. Tāpiri 126 ki te 18.

$$b=\frac{144}{162}$$

Whakahekea te hautanga \frac{144}{162} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 18.

$$b=\frac{8}{9}$$

Nā, me whakaoti te whārite b=\frac{126±18}{162} ina he tango te ±. Tango 18 mai i 126.

$$b=\frac{108}{162}$$

Whakahekea te hautanga \frac{108}{162} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 54.

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

Kua oti te whārite te whakatau.

$$b=\frac{8}{9} b=\frac{2}{3}$$

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.

$$81b^{2}-126b+48=0$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$81b^{2}-126b+48-48=-48$$

Mā te tango i te 48 i a ia ake anō ka toe ko te 0.

$$81b^{2}-126b=-48$$

Whakawehea ngā taha e rua ki te 81.

$$\frac{81b^{2}-126b}{81}=-\frac{48}{81}$$

Mā te whakawehe ki te 81 ka wetekia te whakareanga ki te 81.

$$b^{2}+\left(-\frac{126}{81}\right)b=-\frac{48}{81}$$

Whakahekea te hautanga \frac{-126}{81} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 9.

$$b^{2}-\frac{14}{9}b=-\frac{48}{81}$$

Whakahekea te hautanga \frac{-48}{81} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$b^{2}-\frac{14}{9}b=-\frac{16}{27}$$

Whakawehea te -\frac{14}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{9}. Nā, tāpiria te pūrua o te -\frac{7}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$b^{2}-\frac{14}{9}b+\left(-\frac{7}{9}\right)^{2}=-\frac{16}{27}+\left(-\frac{7}{9}\right)^{2}$$

Pūruatia -\frac{7}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$b^{2}-\frac{14}{9}b+\frac{49}{81}=-\frac{16}{27}+\frac{49}{81}$$

Tāpiri -\frac{16}{27} ki te \frac{49}{81} 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.

$$b^{2}-\frac{14}{9}b+\frac{49}{81}=\frac{1}{81}$$

Tauwehea b^{2}-\frac{14}{9}b+\frac{49}{81}. 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}.

$$\left(b-\frac{7}{9}\right)^{2}=\frac{1}{81}$$

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

$$\sqrt{\left(b-\frac{7}{9}\right)^{2}}=\sqrt{\frac{1}{81}}$$

Whakarūnātia.

$$b-\frac{7}{9}=\frac{1}{9} b-\frac{7}{9}=-\frac{1}{9}$$

Me tāpiri \frac{7}{9} ki ngā taha e rua o te whārite.

$$b=\frac{8}{9} b=\frac{2}{3}$$