Whakaoti i te 3wleft(w+8right)+wleft(w-4right)/2-3=5-w^2

$$\frac{ 3w \left( w+8 \right) +w \left( w-4 \right) }{ 2 } -3 = 5- { w }^{ 2 }$$

Whakaoti mō w

$$w=-4$$

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia ngā taha e rua o te whārite ki te 2.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3w ki te w+8.

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te w ki te w-4.

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

Pahekotia te 3w^{2} me w^{2}, ka 4w^{2}.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

Pahekotia te 24w me -4w, ka 20w.

$$4w^{2}+20w-6=10-2w^{2}$$

Tangohia te 10 mai i ngā taha e rua.

$$4w^{2}+20w-6-10=-2w^{2}$$

Tangohia te 10 i te -6, ka -16.

$$4w^{2}+20w-16=-2w^{2}$$

Me tāpiri te 2w^{2} ki ngā taha e rua.

$$4w^{2}+20w-16+2w^{2}=0$$

Pahekotia te 4w^{2} me 2w^{2}, ka 6w^{2}.

$$6w^{2}+20w-16=0$$

Whakawehea ngā taha e rua ki te 2.

$$3w^{2}+10w-8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3w^{2}+aw+bw-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=3\left(-8\right)=-24$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -24.

$$-1,24 -2,12 -3,8 -4,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-2 b=12$$

Tuhia anō te 3w^{2}+10w-8 hei \left(3w^{2}-2w\right)+\left(12w-8\right).

$$\left(3w^{2}-2w\right)+\left(12w-8\right)$$

Tauwehea te w i te tuatahi me te 4 i te rōpū tuarua.

$$w\left(3w-2\right)+4\left(3w-2\right)$$

Whakatauwehea atu te kīanga pātahi 3w-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3w-2\right)\left(w+4\right)$$

Hei kimi otinga whārite, me whakaoti te 3w-2=0 me te w+4=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3w ki te w+8.

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te w ki te w-4.

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

Pahekotia te 3w^{2} me w^{2}, ka 4w^{2}.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

Pahekotia te 24w me -4w, ka 20w.

$$4w^{2}+20w-6=10-2w^{2}$$

Tangohia te 10 mai i ngā taha e rua.

$$4w^{2}+20w-6-10=-2w^{2}$$

Tangohia te 10 i te -6, ka -16.

$$4w^{2}+20w-16=-2w^{2}$$

Me tāpiri te 2w^{2} ki ngā taha e rua.

$$4w^{2}+20w-16+2w^{2}=0$$

Pahekotia te 4w^{2} me 2w^{2}, ka 6w^{2}.

$$6w^{2}+20w-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 20 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-20±\sqrt{20^{2}-4\times 6\left(-16\right)}}{2\times 6}$$

Pūrua 20.

$$w=\frac{-20±\sqrt{400-4\times 6\left(-16\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$w=\frac{-20±\sqrt{400-24\left(-16\right)}}{2\times 6}$$

Whakareatia -24 ki te -16.

$$w=\frac{-20±\sqrt{400+384}}{2\times 6}$$

Tāpiri 400 ki te 384.

$$w=\frac{-20±\sqrt{784}}{2\times 6}$$

Tuhia te pūtakerua o te 784.

$$w=\frac{-20±28}{2\times 6}$$

Whakareatia 2 ki te 6.

$$w=\frac{-20±28}{12}$$

Nā, me whakaoti te whārite w=\frac{-20±28}{12} ina he tāpiri te ±. Tāpiri -20 ki te 28.

$$w=\frac{8}{12}$$

Whakahekea te hautanga \frac{8}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite w=\frac{-20±28}{12} ina he tango te ±. Tango 28 mai i -20.

$$w=-\frac{48}{12}$$

Whakawehe -48 ki te 12.

$$w=-4$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 2.

$$3w\left(w+8\right)+w\left(w-4\right)-6=10-2w^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3w ki te w+8.

$$3w^{2}+24w+w\left(w-4\right)-6=10-2w^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te w ki te w-4.

$$3w^{2}+24w+w^{2}-4w-6=10-2w^{2}$$

Pahekotia te 3w^{2} me w^{2}, ka 4w^{2}.

$$4w^{2}+24w-4w-6=10-2w^{2}$$

Pahekotia te 24w me -4w, ka 20w.

$$4w^{2}+20w-6=10-2w^{2}$$

Me tāpiri te 2w^{2} ki ngā taha e rua.

$$4w^{2}+20w-6+2w^{2}=10$$

Pahekotia te 4w^{2} me 2w^{2}, ka 6w^{2}.

$$6w^{2}+20w-6=10$$

Me tāpiri te 6 ki ngā taha e rua.

$$6w^{2}+20w=10+6$$

Tāpirihia te 10 ki te 6, ka 16.

$$6w^{2}+20w=16$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6w^{2}+20w}{6}=\frac{16}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$w^{2}+\frac{20}{6}w=\frac{16}{6}$$

Whakahekea te hautanga \frac{20}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$w^{2}+\frac{10}{3}w=\frac{16}{6}$$

Whakahekea te hautanga \frac{16}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$w^{2}+\frac{10}{3}w=\frac{8}{3}$$

Whakawehea te \frac{10}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{3}. Nā, tāpiria te pūrua o te \frac{5}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$w^{2}+\frac{10}{3}w+\left(\frac{5}{3}\right)^{2}=\frac{8}{3}+\left(\frac{5}{3}\right)^{2}$$

Pūruatia \frac{5}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=\frac{8}{3}+\frac{25}{9}$$

Tāpiri \frac{8}{3} ki te \frac{25}{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.

$$w^{2}+\frac{10}{3}w+\frac{25}{9}=\frac{49}{9}$$

Tauwehea w^{2}+\frac{10}{3}w+\frac{25}{9}. 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}.

$$\left(w+\frac{5}{3}\right)^{2}=\frac{49}{9}$$

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

$$\sqrt{\left(w+\frac{5}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Whakarūnātia.

$$w+\frac{5}{3}=\frac{7}{3} w+\frac{5}{3}=-\frac{7}{3}$$

Me tango \frac{5}{3} mai i ngā taha e rua o te whārite.

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