Whakaoti i te sqrt{x-3}+sqrt{2x-2}=2

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

Whakaoti mō x

$$x=3$$

Ngā Upane Otinga

Me tango \sqrt{2x-2} mai i ngā taha e rua o te whārite.

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

Pūruatia ngā taha e rua o te whārite.

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

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

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

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

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

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

$$x-3=4-4\sqrt{2x-2}+2x-2$$

Tangohia te 2 i te 4, ka 2.

$$x-3=2-4\sqrt{2x-2}+2x$$

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

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

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

$$x-3-2-2x=-4\sqrt{2x-2}$$

Tangohia te 2 i te -3, ka -5.

$$x-5-2x=-4\sqrt{2x-2}$$

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

$$-x-5=-4\sqrt{2x-2}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(-x-5\right)^{2}=\left(-4\sqrt{2x-2}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-x-5\right)^{2}.

$$x^{2}+10x+25=\left(-4\sqrt{2x-2}\right)^{2}$$

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

$$x^{2}+10x+25=\left(-4\right)^{2}\left(\sqrt{2x-2}\right)^{2}$$

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

$$x^{2}+10x+25=16\left(\sqrt{2x-2}\right)^{2}$$

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

$$x^{2}+10x+25=16\left(2x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te 2x-2.

$$x^{2}+10x+25=32x-32$$

Tangohia te 32x mai i ngā taha e rua.

$$x^{2}+10x+25-32x=-32$$

Pahekotia te 10x me -32x, ka -22x.

$$x^{2}-22x+25=-32$$

Me tāpiri te 32 ki ngā taha e rua.

$$x^{2}-22x+25+32=0$$

Tāpirihia te 25 ki te 32, ka 57.

$$x^{2}-22x+57=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-22x+57 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-22 ab=57$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 57.

$$-1,-57 -3,-19$$

Tātaihia te tapeke mō ia takirua.

$$-1-57=-58 -3-19=-22$$

Ko te otinga te takirua ka hoatu i te tapeke -22.

$$a=-19 b=-3$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-19\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-19=0 me te x-3=0.

$$x=19 x=3$$

Whakakapia te 19 mō te x i te whārite \sqrt{x-3}+\sqrt{2x-2}=2.

$$\sqrt{19-3}+\sqrt{2\times 19-2}=2$$

Whakarūnātia. Ko te uara x=19 kāore e ngata ana ki te whārite.

$$10=2$$

Whakakapia te 3 mō te x i te whārite \sqrt{x-3}+\sqrt{2x-2}=2.

$$\sqrt{3-3}+\sqrt{2\times 3-2}=2$$

Whakarūnātia. Ko te uara x=3 kua ngata te whārite.

$$2=2$$

Ko te whārite \sqrt{x-3}=-\sqrt{2x-2}+2 he rongoā ahurei.

$$x=3$$