Whakaoti i te 3(w/6)^2-8(w/6)+4=0

$$3 ( \frac { w } { 6 } ) ^ { 2 } - 8 ( \frac { w } { 6 } ) + 4 = 0$$

Whakaoti mō w

$$w=4$$

$$w=12$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 6.

$$6\left(3\times \left(\frac{w}{6}\right)^{2}-8\times \frac{w}{6}\right)+24=0$$

Kia whakarewa i te \frac{w}{6} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$6\left(3\times \frac{w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Tuhia te 3\times \frac{w^{2}}{6^{2}} hei hautanga kotahi.

$$6\left(\frac{3w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Tuhia te 8\times \frac{w}{6} hei hautanga kotahi.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{8w}{6}\right)+24=0$$

Whakawehea te 8w ki te 6, kia riro ko \frac{4}{3}w.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{4}{3}w\right)+24=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te \frac{3w^{2}}{6^{2}}-\frac{4}{3}w.

$$6\times \frac{3w^{2}}{6^{2}}+6\left(-\frac{4}{3}w\right)+24=0$$

Tātaihia te 6 mā te pū o 2, kia riro ko 36.

$$6\times \frac{3w^{2}}{36}+6\left(-\frac{4}{3}w\right)+24=0$$

Whakawehea te 3w^{2} ki te 36, kia riro ko \frac{1}{12}w^{2}.

$$6\times \frac{1}{12}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Whakareatia te 6 ki te \frac{1}{12}, ka \frac{1}{2}.

$$\frac{1}{2}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Whakareatia te 6 ki te -1, ka -6.

$$\frac{1}{2}w^{2}-6\times \frac{4}{3}w+24=0$$

Whakareatia te -6 ki te \frac{4}{3}, ka -8.

$$\frac{1}{2}w^{2}-8w+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{2} mō a, -8 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times \frac{1}{2}\times 24}}{2\times \frac{1}{2}}$$

Pūrua -8.

$$w=\frac{-\left(-8\right)±\sqrt{64-4\times \frac{1}{2}\times 24}}{2\times \frac{1}{2}}$$

Whakareatia -4 ki te \frac{1}{2}.

$$w=\frac{-\left(-8\right)±\sqrt{64-2\times 24}}{2\times \frac{1}{2}}$$

Whakareatia -2 ki te 24.

$$w=\frac{-\left(-8\right)±\sqrt{64-48}}{2\times \frac{1}{2}}$$

Tāpiri 64 ki te -48.

$$w=\frac{-\left(-8\right)±\sqrt{16}}{2\times \frac{1}{2}}$$

Tuhia te pūtakerua o te 16.

$$w=\frac{-\left(-8\right)±4}{2\times \frac{1}{2}}$$

Ko te tauaro o -8 ko 8.

$$w=\frac{8±4}{2\times \frac{1}{2}}$$

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

$$w=\frac{8±4}{1}$$

Nā, me whakaoti te whārite w=\frac{8±4}{1} ina he tāpiri te ±. Tāpiri 8 ki te 4.

$$w=\frac{12}{1}$$

Whakawehe 12 ki te 1.

$$w=12$$

Nā, me whakaoti te whārite w=\frac{8±4}{1} ina he tango te ±. Tango 4 mai i 8.

$$w=\frac{4}{1}$$

Whakawehe 4 ki te 1.

$$w=4$$

Kua oti te whārite te whakatau.

$$w=12 w=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 6.

$$6\left(3\times \left(\frac{w}{6}\right)^{2}-8\times \frac{w}{6}\right)+24=0$$

Kia whakarewa i te \frac{w}{6} ki tētahi taupū, me whakarewa tahi te taurunga me te tauraro ki te taupū kātahi ka whakawehe.

$$6\left(3\times \frac{w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Tuhia te 3\times \frac{w^{2}}{6^{2}} hei hautanga kotahi.

$$6\left(\frac{3w^{2}}{6^{2}}-8\times \frac{w}{6}\right)+24=0$$

Tuhia te 8\times \frac{w}{6} hei hautanga kotahi.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{8w}{6}\right)+24=0$$

Whakawehea te 8w ki te 6, kia riro ko \frac{4}{3}w.

$$6\left(\frac{3w^{2}}{6^{2}}-\frac{4}{3}w\right)+24=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te \frac{3w^{2}}{6^{2}}-\frac{4}{3}w.

$$6\times \frac{3w^{2}}{6^{2}}+6\left(-\frac{4}{3}w\right)+24=0$$

Tātaihia te 6 mā te pū o 2, kia riro ko 36.

$$6\times \frac{3w^{2}}{36}+6\left(-\frac{4}{3}w\right)+24=0$$

Whakawehea te 3w^{2} ki te 36, kia riro ko \frac{1}{12}w^{2}.

$$6\times \frac{1}{12}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Whakareatia te 6 ki te \frac{1}{12}, ka \frac{1}{2}.

$$\frac{1}{2}w^{2}+6\left(-\frac{4}{3}w\right)+24=0$$

Tangohia te 24 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$\frac{1}{2}w^{2}+6\left(-\frac{4}{3}w\right)=-24$$

Whakareatia te 6 ki te -1, ka -6.

$$\frac{1}{2}w^{2}-6\times \frac{4}{3}w=-24$$

Whakareatia te -6 ki te \frac{4}{3}, ka -8.

$$\frac{1}{2}w^{2}-8w=-24$$

Me whakarea ngā taha e rua ki te 2.

$$\frac{\frac{1}{2}w^{2}-8w}{\frac{1}{2}}=-\frac{24}{\frac{1}{2}}$$

Mā te whakawehe ki te \frac{1}{2} ka wetekia te whakareanga ki te \frac{1}{2}.

$$w^{2}+\left(-\frac{8}{\frac{1}{2}}\right)w=-\frac{24}{\frac{1}{2}}$$

Whakawehe -8 ki te \frac{1}{2} mā te whakarea -8 ki te tau huripoki o \frac{1}{2}.

$$w^{2}-16w=-\frac{24}{\frac{1}{2}}$$

Whakawehe -24 ki te \frac{1}{2} mā te whakarea -24 ki te tau huripoki o \frac{1}{2}.

$$w^{2}-16w=-48$$

Whakawehea te -16, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -8. Nā, tāpiria te pūrua o te -8 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$w^{2}-16w+\left(-8\right)^{2}=-48+\left(-8\right)^{2}$$

Pūrua -8.

$$w^{2}-16w+64=-48+64$$

Tāpiri -48 ki te 64.

$$w^{2}-16w+64=16$$

Tauwehea w^{2}-16w+64. 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}.

$$\left(w-8\right)^{2}=16$$

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

$$\sqrt{\left(w-8\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$w-8=4 w-8=-4$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$w=12 w=4$$