Whakaoti i te sqrt{2x-1}-2=sqrt{x-4}

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

Whakaoti mō x

$$x=13$$

$$x=5$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

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

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

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

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

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

Tāpirihia te -1 ki te 4, ka 3.

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

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

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

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

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

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

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

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

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

Tangohia te 3 i te -4, ka -7.

$$-4\sqrt{2x-1}=-x-7$$

Pūruatia ngā taha e rua o te whārite.

$$\left(-4\sqrt{2x-1}\right)^{2}=\left(-x-7\right)^{2}$$

Whakarohaina te \left(-4\sqrt{2x-1}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{2x-1}\right)^{2}=\left(-x-7\right)^{2}$$

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

$$16\left(\sqrt{2x-1}\right)^{2}=\left(-x-7\right)^{2}$$

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

$$16\left(2x-1\right)=\left(-x-7\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te 2x-1.

$$32x-16=\left(-x-7\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(-x-7\right)^{2}.

$$32x-16=x^{2}+14x+49$$

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

$$32x-16-x^{2}=14x+49$$

Tangohia te 14x mai i ngā taha e rua.

$$32x-16-x^{2}-14x=49$$

Pahekotia te 32x me -14x, ka 18x.

$$18x-16-x^{2}=49$$

Tangohia te 49 mai i ngā taha e rua.

$$18x-16-x^{2}-49=0$$

Tangohia te 49 i te -16, ka -65.

$$18x-65-x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+18x-65=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-65. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=18 ab=-\left(-65\right)=65$$

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 65.

$$1,65 5,13$$

Tātaihia te tapeke mō ia takirua.

$$1+65=66 5+13=18$$

Ko te otinga te takirua ka hoatu i te tapeke 18.

$$a=13 b=5$$

Tuhia anō te -x^{2}+18x-65 hei \left(-x^{2}+13x\right)+\left(5x-65\right).

$$\left(-x^{2}+13x\right)+\left(5x-65\right)$$

Tauwehea te -x i te tuatahi me te 5 i te rōpū tuarua.

$$-x\left(x-13\right)+5\left(x-13\right)$$

Whakatauwehea atu te kīanga pātahi x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-13\right)\left(-x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te -x+5=0.

$$x=13 x=5$$

Whakakapia te 13 mō te x i te whārite \sqrt{2x-1}-2=\sqrt{x-4}.

$$\sqrt{2\times 13-1}-2=\sqrt{13-4}$$

Whakarūnātia. Ko te uara x=13 kua ngata te whārite.

$$3=3$$

Whakakapia te 5 mō te x i te whārite \sqrt{2x-1}-2=\sqrt{x-4}.

$$\sqrt{2\times 5-1}-2=\sqrt{5-4}$$

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

$$1=1$$

Rārangihia ngā rongoā katoa o \sqrt{2x-1}-2=\sqrt{x-4}.

$$x=13 x=5$$