Whakaoti i te [2/3(x-3)]^2=16*7-x/2

$$[ \frac { 2 } { 3 } ( x - 3 ) ] ^ { 2 } = 16 \times \frac { 7 - x } { 2 }$$

Whakaoti mō x

$$x=3\sqrt{17}-6\approx 6.369316877$$

$$x=-3\sqrt{17}-6\approx -18.369316877$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$2\times \left(\frac{2}{3}\left(x-3\right)\right)^{2}=16\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{2}{3} ki te x-3.

$$2\left(\frac{2}{3}x-2\right)^{2}=16\left(7-x\right)$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\frac{2}{3}x-2\right)^{2}.

$$2\left(\frac{4}{9}x^{2}-\frac{8}{3}x+4\right)=16\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te \frac{4}{9}x^{2}-\frac{8}{3}x+4.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=16\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te 7-x.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=112-16x$$

Tangohia te 112 mai i ngā taha e rua.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8-112=-16x$$

Tangohia te 112 i te 8, ka -104.

$$\frac{8}{9}x^{2}-\frac{16}{3}x-104=-16x$$

Me tāpiri te 16x ki ngā taha e rua.

$$\frac{8}{9}x^{2}-\frac{16}{3}x-104+16x=0$$

Pahekotia te -\frac{16}{3}x me 16x, ka \frac{32}{3}x.

$$\frac{8}{9}x^{2}+\frac{32}{3}x-104=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{8}{9} mō a, \frac{32}{3} mō b, me -104 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{32}{3}±\sqrt{\left(\frac{32}{3}\right)^{2}-4\times \frac{8}{9}\left(-104\right)}}{2\times \frac{8}{9}}$$

Pūruatia \frac{32}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{1024}{9}-4\times \frac{8}{9}\left(-104\right)}}{2\times \frac{8}{9}}$$

Whakareatia -4 ki te \frac{8}{9}.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{1024}{9}-\frac{32}{9}\left(-104\right)}}{2\times \frac{8}{9}}$$

Whakareatia -\frac{32}{9} ki te -104.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{1024+3328}{9}}}{2\times \frac{8}{9}}$$

Tāpiri \frac{1024}{9} ki te \frac{3328}{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.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{4352}{9}}}{2\times \frac{8}{9}}$$

Tuhia te pūtakerua o te \frac{4352}{9}.

$$x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{2\times \frac{8}{9}}$$

Whakareatia 2 ki te \frac{8}{9}.

$$x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{\frac{16}{9}}$$

Nā, me whakaoti te whārite x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{\frac{16}{9}} ina he tāpiri te ±. Tāpiri -\frac{32}{3} ki te \frac{16\sqrt{17}}{3}.

$$x=\frac{16\sqrt{17}-32}{\frac{16}{9}\times 3}$$

Whakawehe \frac{-32+16\sqrt{17}}{3} ki te \frac{16}{9} mā te whakarea \frac{-32+16\sqrt{17}}{3} ki te tau huripoki o \frac{16}{9}.

$$x=3\sqrt{17}-6$$

Nā, me whakaoti te whārite x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{\frac{16}{9}} ina he tango te ±. Tango \frac{16\sqrt{17}}{3} mai i -\frac{32}{3}.

$$x=\frac{-16\sqrt{17}-32}{\frac{16}{9}\times 3}$$

Whakawehe \frac{-32-16\sqrt{17}}{3} ki te \frac{16}{9} mā te whakarea \frac{-32-16\sqrt{17}}{3} ki te tau huripoki o \frac{16}{9}.

$$x=-3\sqrt{17}-6$$

Kua oti te whārite te whakatau.

$$x=3\sqrt{17}-6 x=-3\sqrt{17}-6$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$2\times \left(\frac{2}{3}\left(x-3\right)\right)^{2}=16\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{2}{3} ki te x-3.

$$2\left(\frac{2}{3}x-2\right)^{2}=16\left(7-x\right)$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\frac{2}{3}x-2\right)^{2}.

$$2\left(\frac{4}{9}x^{2}-\frac{8}{3}x+4\right)=16\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te \frac{4}{9}x^{2}-\frac{8}{3}x+4.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=16\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te 7-x.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=112-16x$$

Me tāpiri te 16x ki ngā taha e rua.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8+16x=112$$

Pahekotia te -\frac{16}{3}x me 16x, ka \frac{32}{3}x.

$$\frac{8}{9}x^{2}+\frac{32}{3}x+8=112$$

Tangohia te 8 mai i ngā taha e rua.

$$\frac{8}{9}x^{2}+\frac{32}{3}x=112-8$$

Tangohia te 8 i te 112, ka 104.

$$\frac{8}{9}x^{2}+\frac{32}{3}x=104$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{9}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{8}{9}x^{2}+\frac{32}{3}x}{\frac{8}{9}}=\frac{104}{\frac{8}{9}}$$

Mā te whakawehe ki te \frac{8}{9} ka wetekia te whakareanga ki te \frac{8}{9}.

$$x^{2}+\frac{\frac{32}{3}}{\frac{8}{9}}x=\frac{104}{\frac{8}{9}}$$

Whakawehe \frac{32}{3} ki te \frac{8}{9} mā te whakarea \frac{32}{3} ki te tau huripoki o \frac{8}{9}.

$$x^{2}+12x=\frac{104}{\frac{8}{9}}$$

Whakawehe 104 ki te \frac{8}{9} mā te whakarea 104 ki te tau huripoki o \frac{8}{9}.

$$x^{2}+12x=117$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=117+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=117+36$$

Tāpiri 117 ki te 36.

$$x^{2}+12x+36=153$$

Tauwehea x^{2}+12x+36. 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}.

$$\left(x+6\right)^{2}=153$$

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

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{153}$$

Whakarūnātia.

$$x+6=3\sqrt{17} x+6=-3\sqrt{17}$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=3\sqrt{17}-6 x=-3\sqrt{17}-6$$