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

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

Whakaoti mō x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Ngā Upane Otinga

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

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

Tangohia te 2 i te -1, ka -3.

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

Pūruatia ngā taha e rua o te whārite.

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

Tātaihia te \sqrt{3x-1} mā te pū o 2, kia riro ko 3x-1.

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

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

$$3x-1=9x^{2}-18x+9$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$3x-1-9x^{2}=-18x+9$$

Me tāpiri te 18x ki ngā taha e rua.

$$3x-1-9x^{2}+18x=9$$

Pahekotia te 3x me 18x, ka 21x.

$$21x-1-9x^{2}=9$$

Tangohia te 9 mai i ngā taha e rua.

$$21x-1-9x^{2}-9=0$$

Tangohia te 9 i te -1, ka -10.

$$21x-10-9x^{2}=0$$

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

$$-9x^{2}+21x-10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -9x^{2}+ax+bx-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=21 ab=-9\left(-10\right)=90$$

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 90.

$$1,90 2,45 3,30 5,18 6,15 9,10$$

Tātaihia te tapeke mō ia takirua.

$$1+90=91 2+45=47 3+30=33 5+18=23 6+15=21 9+10=19$$

Ko te otinga te takirua ka hoatu i te tapeke 21.

$$a=15 b=6$$

Tuhia anō te -9x^{2}+21x-10 hei \left(-9x^{2}+15x\right)+\left(6x-10\right).

$$\left(-9x^{2}+15x\right)+\left(6x-10\right)$$

Tauwehea te -3x i te tuatahi me te 2 i te rōpū tuarua.

$$-3x\left(3x-5\right)+2\left(3x-5\right)$$

Whakatauwehea atu te kīanga pātahi 3x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-5\right)\left(-3x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-5=0 me te -3x+2=0.

$$x=\frac{5}{3} x=\frac{2}{3}$$

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

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

Whakarūnātia. Ko te uara x=\frac{5}{3} kua ngata te whārite.

$$4=4$$

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

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

Whakarūnātia. Ko te uara x=\frac{2}{3} kāore e ngata ana ki te whārite.

$$3=1$$

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

$$x=\frac{5}{3}$$