Whakaoti i te 3w^2-12w+7=0

$$3 w ^ { 2 } - 12 w + 7 = 0$$

Whakaoti mō w

$$w=\frac{\sqrt{15}}{3}+2\approx 3.290994449$$

$$w=-\frac{\sqrt{15}}{3}+2\approx 0.709005551$$

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.

$$3w^{2}-12w+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -12 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 3\times 7}}{2\times 3}$$

Pūrua -12.

$$w=\frac{-\left(-12\right)±\sqrt{144-4\times 3\times 7}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$w=\frac{-\left(-12\right)±\sqrt{144-12\times 7}}{2\times 3}$$

Whakareatia -12 ki te 7.

$$w=\frac{-\left(-12\right)±\sqrt{144-84}}{2\times 3}$$

Tāpiri 144 ki te -84.

$$w=\frac{-\left(-12\right)±\sqrt{60}}{2\times 3}$$

Tuhia te pūtakerua o te 60.

$$w=\frac{-\left(-12\right)±2\sqrt{15}}{2\times 3}$$

Ko te tauaro o -12 ko 12.

$$w=\frac{12±2\sqrt{15}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$w=\frac{12±2\sqrt{15}}{6}$$

Nā, me whakaoti te whārite w=\frac{12±2\sqrt{15}}{6} ina he tāpiri te ±. Tāpiri 12 ki te 2\sqrt{15}.

$$w=\frac{2\sqrt{15}+12}{6}$$

Whakawehe 12+2\sqrt{15} ki te 6.

$$w=\frac{\sqrt{15}}{3}+2$$

Nā, me whakaoti te whārite w=\frac{12±2\sqrt{15}}{6} ina he tango te ±. Tango 2\sqrt{15} mai i 12.

$$w=\frac{12-2\sqrt{15}}{6}$$

Whakawehe 12-2\sqrt{15} ki te 6.

$$w=-\frac{\sqrt{15}}{3}+2$$

Kua oti te whārite te whakatau.

$$w=\frac{\sqrt{15}}{3}+2 w=-\frac{\sqrt{15}}{3}+2$$

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.

$$3w^{2}-12w+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$3w^{2}-12w+7-7=-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$3w^{2}-12w=-7$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3w^{2}-12w}{3}=-\frac{7}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$w^{2}+\left(-\frac{12}{3}\right)w=-\frac{7}{3}$$

Whakawehe -12 ki te 3.

$$w^{2}-4w=-\frac{7}{3}$$

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

$$w^{2}-4w+\left(-2\right)^{2}=-\frac{7}{3}+\left(-2\right)^{2}$$

Pūrua -2.

$$w^{2}-4w+4=-\frac{7}{3}+4$$

Tāpiri -\frac{7}{3} ki te 4.

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

Tauwehea w^{2}-4w+4. 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}.

$$\left(w-2\right)^{2}=\frac{5}{3}$$

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

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

Whakarūnātia.

$$w-2=\frac{\sqrt{15}}{3} w-2=-\frac{\sqrt{15}}{3}$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$w=\frac{\sqrt{15}}{3}+2 w=-\frac{\sqrt{15}}{3}+2$$