Whakaoti i te sqrt{2x-1}-sqrt{x-1}=sqrt{6-x}

$$\sqrt { 2 x - 1 } - \sqrt { x - 1 } = \sqrt { 6 - x }$$

Whakaoti mō x

$$x=5$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{2x-1}-\sqrt{x-1}\right)^{2}=\left(\sqrt{6-x}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{2x-1}-\sqrt{x-1}\right)^{2}.

$$\left(\sqrt{2x-1}\right)^{2}-2\sqrt{2x-1}\sqrt{x-1}+\left(\sqrt{x-1}\right)^{2}=\left(\sqrt{6-x}\right)^{2}$$

Tātaihia te \sqrt{2x-1} mā te pū o 2, kia riro ko 2x-1.

$$2x-1-2\sqrt{2x-1}\sqrt{x-1}+\left(\sqrt{x-1}\right)^{2}=\left(\sqrt{6-x}\right)^{2}$$

Tātaihia te \sqrt{x-1} mā te pū o 2, kia riro ko x-1.

$$2x-1-2\sqrt{2x-1}\sqrt{x-1}+x-1=\left(\sqrt{6-x}\right)^{2}$$

Pahekotia te 2x me x, ka 3x.

$$3x-1-2\sqrt{2x-1}\sqrt{x-1}-1=\left(\sqrt{6-x}\right)^{2}$$

Tangohia te 1 i te -1, ka -2.

$$3x-2-2\sqrt{2x-1}\sqrt{x-1}=\left(\sqrt{6-x}\right)^{2}$$

Tātaihia te \sqrt{6-x} mā te pū o 2, kia riro ko 6-x.

$$3x-2-2\sqrt{2x-1}\sqrt{x-1}=6-x$$

Me tango 3x-2 mai i ngā taha e rua o te whārite.

$$-2\sqrt{2x-1}\sqrt{x-1}=6-x-\left(3x-2\right)$$

Hei kimi i te tauaro o 3x-2, kimihia te tauaro o ia taurangi.

$$-2\sqrt{2x-1}\sqrt{x-1}=6-x-3x+2$$

Pahekotia te -x me -3x, ka -4x.

$$-2\sqrt{2x-1}\sqrt{x-1}=6-4x+2$$

Tāpirihia te 6 ki te 2, ka 8.

$$-2\sqrt{2x-1}\sqrt{x-1}=8-4x$$

Pūruatia ngā taha e rua o te whārite.

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

Whakarohaina te \left(-2\sqrt{2x-1}\sqrt{x-1}\right)^{2}.

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

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

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

Tātaihia te \sqrt{2x-1} mā te pū o 2, kia riro ko 2x-1.

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

Tātaihia te \sqrt{x-1} mā te pū o 2, kia riro ko x-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-1.

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

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 8x-4 ki ia tau o x-1.

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

Pahekotia te -8x me -4x, ka -12x.

$$8x^{2}-12x+4=\left(8-4x\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(8-4x\right)^{2}.

$$8x^{2}-12x+4=64-64x+16x^{2}$$

Tangohia te 64 mai i ngā taha e rua.

$$8x^{2}-12x+4-64=-64x+16x^{2}$$

Tangohia te 64 i te 4, ka -60.

$$8x^{2}-12x-60=-64x+16x^{2}$$

Me tāpiri te 64x ki ngā taha e rua.

$$8x^{2}-12x-60+64x=16x^{2}$$

Pahekotia te -12x me 64x, ka 52x.

$$8x^{2}+52x-60=16x^{2}$$

Tangohia te 16x^{2} mai i ngā taha e rua.

$$8x^{2}+52x-60-16x^{2}=0$$

Pahekotia te 8x^{2} me -16x^{2}, ka -8x^{2}.

$$-8x^{2}+52x-60=0$$

Whakawehea ngā taha e rua ki te 4.

$$-2x^{2}+13x-15=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -2x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=-2\left(-15\right)=30$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 30.

$$1,30 2,15 3,10 5,6$$

Tātaihia te tapeke mō ia takirua.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=10 b=3$$

Tuhia anō te -2x^{2}+13x-15 hei \left(-2x^{2}+10x\right)+\left(3x-15\right).

$$\left(-2x^{2}+10x\right)+\left(3x-15\right)$$

Tauwehea te 2x i te tuatahi me te -3 i te rōpū tuarua.

$$2x\left(-x+5\right)-3\left(-x+5\right)$$

Whakatauwehea atu te kīanga pātahi -x+5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+5\right)\left(2x-3\right)$$

Hei kimi otinga whārite, me whakaoti te -x+5=0 me te 2x-3=0.

$$x=5 x=\frac{3}{2}$$

Whakakapia te 5 mō te x i te whārite \sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x}.

$$\sqrt{2\times 5-1}-\sqrt{5-1}=\sqrt{6-5}$$

Whakarūnātia. Ko te uara x=5 kua ngata te whārite.

$$1=1$$

Whakakapia te \frac{3}{2} mō te x i te whārite \sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x}.

$$\sqrt{2\times \frac{3}{2}-1}-\sqrt{\frac{3}{2}-1}=\sqrt{6-\frac{3}{2}}$$

Whakarūnātia. Ko te uara x=\frac{3}{2} kāore e ngata ana ki te whārite.

$$\frac{1}{2}\times 2^{\frac{1}{2}}=\frac{3}{2}\times 2^{\frac{1}{2}}$$

Whakakapia te 5 mō te x i te whārite \sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x}.

$$\sqrt{2\times 5-1}-\sqrt{5-1}=\sqrt{6-5}$$

Whakarūnātia. Ko te uara x=5 kua ngata te whārite.

$$1=1$$

Ko te whārite \sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x} he rongoā ahurei.

$$x=5$$