Whakaoti i te sqrt{3x+4}-sqrt{x-4}=4

$$\sqrt { 3 x + 4 } - \sqrt { x - 4 } = 4$$

Whakaoti mō x

$$x=20$$

$$x=4$$

Ngā Upane Otinga

Me tango -\sqrt{x-4} mai i ngā taha e rua o te whārite.

$$\sqrt{3x+4}=4+\sqrt{x-4}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{3x+4}\right)^{2}=\left(4+\sqrt{x-4}\right)^{2}$$

Tātaihia te \sqrt{3x+4} mā te pū o 2, kia riro ko 3x+4.

$$3x+4=\left(4+\sqrt{x-4}\right)^{2}$$

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

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

Tātaihia te \sqrt{x-4} mā te pū o 2, kia riro ko x-4.

$$3x+4=16+8\sqrt{x-4}+x-4$$

Tangohia te 4 i te 16, ka 12.

$$3x+4=12+8\sqrt{x-4}+x$$

Me tango 12+x mai i ngā taha e rua o te whārite.

$$3x+4-\left(12+x\right)=8\sqrt{x-4}$$

Hei kimi i te tauaro o 12+x, kimihia te tauaro o ia taurangi.

$$3x+4-12-x=8\sqrt{x-4}$$

Tangohia te 12 i te 4, ka -8.

$$3x-8-x=8\sqrt{x-4}$$

Pahekotia te 3x me -x, ka 2x.

$$2x-8=8\sqrt{x-4}$$

Pūruatia ngā taha e rua o te whārite.

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

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

$$4x^{2}-32x+64=\left(8\sqrt{x-4}\right)^{2}$$

Whakarohaina te \left(8\sqrt{x-4}\right)^{2}.

$$4x^{2}-32x+64=8^{2}\left(\sqrt{x-4}\right)^{2}$$

Tātaihia te 8 mā te pū o 2, kia riro ko 64.

$$4x^{2}-32x+64=64\left(\sqrt{x-4}\right)^{2}$$

Tātaihia te \sqrt{x-4} mā te pū o 2, kia riro ko x-4.

$$4x^{2}-32x+64=64\left(x-4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 64 ki te x-4.

$$4x^{2}-32x+64=64x-256$$

Tangohia te 64x mai i ngā taha e rua.

$$4x^{2}-32x+64-64x=-256$$

Pahekotia te -32x me -64x, ka -96x.

$$4x^{2}-96x+64=-256$$

Me tāpiri te 256 ki ngā taha e rua.

$$4x^{2}-96x+64+256=0$$

Tāpirihia te 64 ki te 256, ka 320.

$$4x^{2}-96x+320=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -96 mō b, me 320 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-96\right)±\sqrt{\left(-96\right)^{2}-4\times 4\times 320}}{2\times 4}$$

Pūrua -96.

$$x=\frac{-\left(-96\right)±\sqrt{9216-4\times 4\times 320}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-96\right)±\sqrt{9216-16\times 320}}{2\times 4}$$

Whakareatia -16 ki te 320.

$$x=\frac{-\left(-96\right)±\sqrt{9216-5120}}{2\times 4}$$

Tāpiri 9216 ki te -5120.

$$x=\frac{-\left(-96\right)±\sqrt{4096}}{2\times 4}$$

Tuhia te pūtakerua o te 4096.

$$x=\frac{-\left(-96\right)±64}{2\times 4}$$

Ko te tauaro o -96 ko 96.

$$x=\frac{96±64}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{96±64}{8}$$

Nā, me whakaoti te whārite x=\frac{96±64}{8} ina he tāpiri te ±. Tāpiri 96 ki te 64.

$$x=\frac{160}{8}$$

Whakawehe 160 ki te 8.

$$x=20$$

Nā, me whakaoti te whārite x=\frac{96±64}{8} ina he tango te ±. Tango 64 mai i 96.

$$x=\frac{32}{8}$$

Whakawehe 32 ki te 8.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=20 x=4$$

Whakakapia te 20 mō te x i te whārite \sqrt{3x+4}-\sqrt{x-4}=4.

$$\sqrt{3\times 20+4}-\sqrt{20-4}=4$$

Whakarūnātia. Ko te uara x=20 kua ngata te whārite.

$$4=4$$

Whakakapia te 4 mō te x i te whārite \sqrt{3x+4}-\sqrt{x-4}=4.

$$\sqrt{3\times 4+4}-\sqrt{4-4}=4$$

Whakarūnātia. Ko te uara x=4 kua ngata te whārite.

$$4=4$$

Rārangihia ngā rongoā katoa o \sqrt{3x+4}=\sqrt{x-4}+4.

$$x=20 x=4$$