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

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

Whakaoti mō x

$$x=14$$

$$x=6$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

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

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

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

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

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

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

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

Tangohia te 5 i te 4, ka -1.

$$2x-3=-1+4\sqrt{x-5}+x$$

Me tango -1+x mai i ngā taha e rua o te whārite.

$$2x-3-\left(-1+x\right)=4\sqrt{x-5}$$

Hei kimi i te tauaro o -1+x, kimihia te tauaro o ia taurangi.

$$2x-3+1-x=4\sqrt{x-5}$$

Tāpirihia te -3 ki te 1, ka -2.

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

Pahekotia te 2x me -x, ka x.

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

Pūruatia ngā taha e rua o te whārite.

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

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

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

Whakarohaina te \left(4\sqrt{x-5}\right)^{2}.

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

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

$$x^{2}-4x+4=16\left(\sqrt{x-5}\right)^{2}$$

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

$$x^{2}-4x+4=16\left(x-5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te x-5.

$$x^{2}-4x+4=16x-80$$

Tangohia te 16x mai i ngā taha e rua.

$$x^{2}-4x+4-16x=-80$$

Pahekotia te -4x me -16x, ka -20x.

$$x^{2}-20x+4=-80$$

Me tāpiri te 80 ki ngā taha e rua.

$$x^{2}-20x+4+80=0$$

Tāpirihia te 4 ki te 80, ka 84.

$$x^{2}-20x+84=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-20x+84 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.

$$a+b=-20 ab=84$$

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 84.

$$-1,-84 -2,-42 -3,-28 -4,-21 -6,-14 -7,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-84=-85 -2-42=-44 -3-28=-31 -4-21=-25 -6-14=-20 -7-12=-19$$

Ko te otinga te takirua ka hoatu i te tapeke -20.

$$a=-14 b=-6$$

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

$$\left(x-14\right)\left(x-6\right)$$

Hei kimi otinga whārite, me whakaoti te x-14=0 me te x-6=0.

$$x=14 x=6$$

Whakakapia te 14 mō te x i te whārite \sqrt{2x-3}=2+\sqrt{x-5}.

$$\sqrt{2\times 14-3}=2+\sqrt{14-5}$$

Whakarūnātia. Ko te uara x=14 kua ngata te whārite.

$$5=5$$

Whakakapia te 6 mō te x i te whārite \sqrt{2x-3}=2+\sqrt{x-5}.

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

Whakarūnātia. Ko te uara x=6 kua ngata te whārite.

$$3=3$$

Rārangihia ngā rongoā katoa o \sqrt{2x-3}=\sqrt{x-5}+2.

$$x=14 x=6$$