Whakaoti i te sqrt{6+sqrt{x+4}}=sqrt{2x-1}

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

Whakaoti mō x

$$x=5$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{6+\sqrt{x+4}}\right)^{2}=\left(\sqrt{2x-1}\right)^{2}$$

Tātaihia te \sqrt{6+\sqrt{x+4}} mā te pū o 2, kia riro ko 6+\sqrt{x+4}.

$$6+\sqrt{x+4}=\left(\sqrt{2x-1}\right)^{2}$$

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

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

Me tango 6 mai i ngā taha e rua o te whārite.

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

Tangohia te 6 i te -1, ka -7.

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

Pūruatia ngā taha e rua o te whārite.

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

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

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

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

$$x+4=4x^{2}-28x+49$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$x+4-4x^{2}=-28x+49$$

Me tāpiri te 28x ki ngā taha e rua.

$$x+4-4x^{2}+28x=49$$

Pahekotia te x me 28x, ka 29x.

$$29x+4-4x^{2}=49$$

Tangohia te 49 mai i ngā taha e rua.

$$29x+4-4x^{2}-49=0$$

Tangohia te 49 i te 4, ka -45.

$$29x-45-4x^{2}=0$$

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

$$-4x^{2}+29x-45=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -4x^{2}+ax+bx-45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=29 ab=-4\left(-45\right)=180$$

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 180.

$$1,180 2,90 3,60 4,45 5,36 6,30 9,20 10,18 12,15$$

Tātaihia te tapeke mō ia takirua.

$$1+180=181 2+90=92 3+60=63 4+45=49 5+36=41 6+30=36 9+20=29 10+18=28 12+15=27$$

Ko te otinga te takirua ka hoatu i te tapeke 29.

$$a=20 b=9$$

Tuhia anō te -4x^{2}+29x-45 hei \left(-4x^{2}+20x\right)+\left(9x-45\right).

$$\left(-4x^{2}+20x\right)+\left(9x-45\right)$$

Tauwehea te 4x i te tuatahi me te -9 i te rōpū tuarua.

$$4x\left(-x+5\right)-9\left(-x+5\right)$$

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

$$\left(-x+5\right)\left(4x-9\right)$$

Hei kimi otinga whārite, me whakaoti te -x+5=0 me te 4x-9=0.

$$x=5 x=\frac{9}{4}$$

Whakakapia te 5 mō te x i te whārite \sqrt{6+\sqrt{x+4}}=\sqrt{2x-1}.

$$\sqrt{6+\sqrt{5+4}}=\sqrt{2\times 5-1}$$

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

$$3=3$$

Whakakapia te \frac{9}{4} mō te x i te whārite \sqrt{6+\sqrt{x+4}}=\sqrt{2x-1}.

$$\sqrt{6+\sqrt{\frac{9}{4}+4}}=\sqrt{2\times \frac{9}{4}-1}$$

Whakarūnātia. Ko te uara x=\frac{9}{4} kāore e ngata ana ki te whārite.

$$\frac{1}{2}\times 34^{\frac{1}{2}}=\frac{1}{2}\times 14^{\frac{1}{2}}$$

Whakakapia te 5 mō te x i te whārite \sqrt{6+\sqrt{x+4}}=\sqrt{2x-1}.

$$\sqrt{6+\sqrt{5+4}}=\sqrt{2\times 5-1}$$

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

$$3=3$$

Ko te whārite \sqrt{\sqrt{x+4}+6}=\sqrt{2x-1} he rongoā ahurei.

$$x=5$$