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

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

Whakaoti mō x

$$x=5$$

$$x=1$$

Ngā Upane Otinga

Me tango -\sqrt{2x-1} mai i ngā taha e rua o te whārite.

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

Pūruatia ngā taha e rua o te whārite.

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

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

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

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

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

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

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

Tangohia te 1 i te 1, ka 0.

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

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

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

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

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

Pūruatia ngā taha e rua o te whārite.

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

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

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

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

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

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

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

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x-1.

$$x^{2}+2x+1=8x-4$$

Tangohia te 8x mai i ngā taha e rua.

$$x^{2}+2x+1-8x=-4$$

Pahekotia te 2x me -8x, ka -6x.

$$x^{2}-6x+1=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$x^{2}-6x+1+4=0$$

Tāpirihia te 1 ki te 4, ka 5.

$$x^{2}-6x+5=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-6x+5 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.

$$a+b=-6 ab=5$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-5 b=-1$$

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

$$\left(x-5\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x-1=0.

$$x=5 x=1$$

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

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

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

$$1=1$$

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

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

Whakarūnātia. Ko te uara x=1 kua ngata te whārite.

$$1=1$$

Rārangihia ngā rongoā katoa o \sqrt{3x+1}=\sqrt{2x-1}+1.

$$x=5 x=1$$