Whakaoti i te sqrt{x+2}-sqrt{3x-5}=-1

$$\sqrt { x + 2 } - \sqrt { 3 x - 5 } = - 1$$

Whakaoti mō x

$$x=7$$

Ngā Upane Otinga

Me tango -\sqrt{3x-5} mai i ngā taha e rua o te whārite.

$$\sqrt{x+2}=-1+\sqrt{3x-5}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{x+2}\right)^{2}=\left(-1+\sqrt{3x-5}\right)^{2}$$

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

$$x+2=\left(-1+\sqrt{3x-5}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(-1+\sqrt{3x-5}\right)^{2}.

$$x+2=1-2\sqrt{3x-5}+\left(\sqrt{3x-5}\right)^{2}$$

Tātaihia te \sqrt{3x-5} mā te pū o 2, kia riro ko 3x-5.

$$x+2=1-2\sqrt{3x-5}+3x-5$$

Tangohia te 5 i te 1, ka -4.

$$x+2=-4-2\sqrt{3x-5}+3x$$

Me tango -4+3x mai i ngā taha e rua o te whārite.

$$x+2-\left(-4+3x\right)=-2\sqrt{3x-5}$$

Hei kimi i te tauaro o -4+3x, kimihia te tauaro o ia taurangi.

$$x+2+4-3x=-2\sqrt{3x-5}$$

Tāpirihia te 2 ki te 4, ka 6.

$$x+6-3x=-2\sqrt{3x-5}$$

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

$$-2x+6=-2\sqrt{3x-5}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(-2x+6\right)^{2}=\left(-2\sqrt{3x-5}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(-2x+6\right)^{2}.

$$4x^{2}-24x+36=\left(-2\sqrt{3x-5}\right)^{2}$$

Whakarohaina te \left(-2\sqrt{3x-5}\right)^{2}.

$$4x^{2}-24x+36=\left(-2\right)^{2}\left(\sqrt{3x-5}\right)^{2}$$

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

$$4x^{2}-24x+36=4\left(\sqrt{3x-5}\right)^{2}$$

Tātaihia te \sqrt{3x-5} mā te pū o 2, kia riro ko 3x-5.

$$4x^{2}-24x+36=4\left(3x-5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 3x-5.

$$4x^{2}-24x+36=12x-20$$

Tangohia te 12x mai i ngā taha e rua.

$$4x^{2}-24x+36-12x=-20$$

Pahekotia te -24x me -12x, ka -36x.

$$4x^{2}-36x+36=-20$$

Me tāpiri te 20 ki ngā taha e rua.

$$4x^{2}-36x+36+20=0$$

Tāpirihia te 36 ki te 20, ka 56.

$$4x^{2}-36x+56=0$$

Whakawehea ngā taha e rua ki te 4.

$$x^{2}-9x+14=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+14. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=1\times 14=14$$

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 14.

$$-1,-14 -2,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-14=-15 -2-7=-9$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-7 b=-2$$

Tuhia anō te x^{2}-9x+14 hei \left(x^{2}-7x\right)+\left(-2x+14\right).

$$\left(x^{2}-7x\right)+\left(-2x+14\right)$$

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

$$x\left(x-7\right)-2\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x-2=0.

$$x=7 x=2$$

Whakakapia te 7 mō te x i te whārite \sqrt{x+2}-\sqrt{3x-5}=-1.

$$\sqrt{7+2}-\sqrt{3\times 7-5}=-1$$

Whakarūnātia. Ko te uara x=7 kua ngata te whārite.

$$-1=-1$$

Whakakapia te 2 mō te x i te whārite \sqrt{x+2}-\sqrt{3x-5}=-1.

$$\sqrt{2+2}-\sqrt{3\times 2-5}=-1$$

Whakarūnātia. Ko te uara x=2 kāore e ngata ana ki te whārite nā te mea e rerekē ngā tohu o te taha maui me te taha katau.

$$1=-1$$

Whakakapia te 7 mō te x i te whārite \sqrt{x+2}-\sqrt{3x-5}=-1.

$$\sqrt{7+2}-\sqrt{3\times 7-5}=-1$$

Whakarūnātia. Ko te uara x=7 kua ngata te whārite.

$$-1=-1$$

Ko te whārite \sqrt{x+2}=\sqrt{3x-5}-1 he rongoā ahurei.

$$x=7$$