Whakaoti i te 1/4left(4^2+left(8-xright)^2+left(4+xright)^2right)=88

$$\frac{ 1 }{ 4 } \left( { 4 }^{ 2 } + { \left(8-x \right) }^{ 2 } + { \left(4+x \right) }^{ 2 } \right) = 88$$

Whakaoti mō x

$$x=2\sqrt{33}+2\approx 13.489125293$$

$$x=2-2\sqrt{33}\approx -9.489125293$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua ki te 4, te tau utu o \frac{1}{4}.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=88\times 4$$

Whakareatia te 88 ki te 4, ka 352.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$16+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(8-x\right)^{2}.

$$16+64-16x+x^{2}+\left(4+x\right)^{2}=352$$

Tāpirihia te 16 ki te 64, ka 80.

$$80-16x+x^{2}+\left(4+x\right)^{2}=352$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4+x\right)^{2}.

$$80-16x+x^{2}+16+8x+x^{2}=352$$

Tāpirihia te 80 ki te 16, ka 96.

$$96-16x+x^{2}+8x+x^{2}=352$$

Pahekotia te -16x me 8x, ka -8x.

$$96-8x+x^{2}+x^{2}=352$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$96-8x+2x^{2}=352$$

Tangohia te 352 mai i ngā taha e rua.

$$96-8x+2x^{2}-352=0$$

Tangohia te 352 i te 96, ka -256.

$$-256-8x+2x^{2}=0$$

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.

$$2x^{2}-8x-256=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -8 mō b, me -256 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-256\right)}}{2\times 2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-256\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\left(-256\right)}}{2\times 2}$$

Whakareatia -8 ki te -256.

$$x=\frac{-\left(-8\right)±\sqrt{64+2048}}{2\times 2}$$

Tāpiri 64 ki te 2048.

$$x=\frac{-\left(-8\right)±\sqrt{2112}}{2\times 2}$$

Tuhia te pūtakerua o te 2112.

$$x=\frac{-\left(-8\right)±8\sqrt{33}}{2\times 2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±8\sqrt{33}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{8±8\sqrt{33}}{4}$$

Nā, me whakaoti te whārite x=\frac{8±8\sqrt{33}}{4} ina he tāpiri te ±. Tāpiri 8 ki te 8\sqrt{33}.

$$x=\frac{8\sqrt{33}+8}{4}$$

Whakawehe 8+8\sqrt{33} ki te 4.

$$x=2\sqrt{33}+2$$

Nā, me whakaoti te whārite x=\frac{8±8\sqrt{33}}{4} ina he tango te ±. Tango 8\sqrt{33} mai i 8.

$$x=\frac{8-8\sqrt{33}}{4}$$

Whakawehe 8-8\sqrt{33} ki te 4.

$$x=2-2\sqrt{33}$$

Kua oti te whārite te whakatau.

$$x=2\sqrt{33}+2 x=2-2\sqrt{33}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua ki te 4, te tau utu o \frac{1}{4}.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=88\times 4$$

Whakareatia te 88 ki te 4, ka 352.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$16+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(8-x\right)^{2}.

$$16+64-16x+x^{2}+\left(4+x\right)^{2}=352$$

Tāpirihia te 16 ki te 64, ka 80.

$$80-16x+x^{2}+\left(4+x\right)^{2}=352$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(4+x\right)^{2}.

$$80-16x+x^{2}+16+8x+x^{2}=352$$

Tāpirihia te 80 ki te 16, ka 96.

$$96-16x+x^{2}+8x+x^{2}=352$$

Pahekotia te -16x me 8x, ka -8x.

$$96-8x+x^{2}+x^{2}=352$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$96-8x+2x^{2}=352$$

Tangohia te 96 mai i ngā taha e rua.

$$-8x+2x^{2}=352-96$$

Tangohia te 96 i te 352, ka 256.

$$-8x+2x^{2}=256$$

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.

$$2x^{2}-8x=256$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-8x}{2}=\frac{256}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{8}{2}\right)x=\frac{256}{2}$$

Whakawehe -8 ki te 2.

$$x^{2}-4x=\frac{256}{2}$$

Whakawehe 256 ki te 2.

$$x^{2}-4x=128$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=128+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=128+4$$

Tāpiri 128 ki te 4.

$$x^{2}-4x+4=132$$

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

$$\left(x-2\right)^{2}=132$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{132}$$

Whakarūnātia.

$$x-2=2\sqrt{33} x-2=-2\sqrt{33}$$

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

$$x=2\sqrt{33}+2 x=2-2\sqrt{33}$$