Whakaoti i te sqrt{4y+1}-sqrt{y-2}=3

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

Whakaoti mō y

$$y=6$$

$$y=2$$

Ngā Upane Otinga

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

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

Pūruatia ngā taha e rua o te whārite.

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

Tātaihia te \sqrt{4y+1} mā te pū o 2, kia riro ko 4y+1.

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

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

$$4y+1=9+6\sqrt{y-2}+\left(\sqrt{y-2}\right)^{2}$$

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

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

Tangohia te 2 i te 9, ka 7.

$$4y+1=7+6\sqrt{y-2}+y$$

Me tango 7+y mai i ngā taha e rua o te whārite.

$$4y+1-\left(7+y\right)=6\sqrt{y-2}$$

Hei kimi i te tauaro o 7+y, kimihia te tauaro o ia taurangi.

$$4y+1-7-y=6\sqrt{y-2}$$

Tangohia te 7 i te 1, ka -6.

$$4y-6-y=6\sqrt{y-2}$$

Pahekotia te 4y me -y, ka 3y.

$$3y-6=6\sqrt{y-2}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(3y-6\right)^{2}=\left(6\sqrt{y-2}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3y-6\right)^{2}.

$$9y^{2}-36y+36=\left(6\sqrt{y-2}\right)^{2}$$

Whakarohaina te \left(6\sqrt{y-2}\right)^{2}.

$$9y^{2}-36y+36=6^{2}\left(\sqrt{y-2}\right)^{2}$$

Tātaihia te 6 mā te pū o 2, kia riro ko 36.

$$9y^{2}-36y+36=36\left(\sqrt{y-2}\right)^{2}$$

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

$$9y^{2}-36y+36=36\left(y-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 36 ki te y-2.

$$9y^{2}-36y+36=36y-72$$

Tangohia te 36y mai i ngā taha e rua.

$$9y^{2}-36y+36-36y=-72$$

Pahekotia te -36y me -36y, ka -72y.

$$9y^{2}-72y+36=-72$$

Me tāpiri te 72 ki ngā taha e rua.

$$9y^{2}-72y+36+72=0$$

Tāpirihia te 36 ki te 72, ka 108.

$$9y^{2}-72y+108=0$$

Whakawehea ngā taha e rua ki te 9.

$$y^{2}-8y+12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei y^{2}+ay+by+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\times 12=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-6 b=-2$$

Tuhia anō te y^{2}-8y+12 hei \left(y^{2}-6y\right)+\left(-2y+12\right).

$$\left(y^{2}-6y\right)+\left(-2y+12\right)$$

Tauwehea te y i te tuatahi me te -2 i te rōpū tuarua.

$$y\left(y-6\right)-2\left(y-6\right)$$

Whakatauwehea atu te kīanga pātahi y-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-6\right)\left(y-2\right)$$

Hei kimi otinga whārite, me whakaoti te y-6=0 me te y-2=0.

$$y=6 y=2$$

Whakakapia te 6 mō te y i te whārite \sqrt{4y+1}-\sqrt{y-2}=3.

$$\sqrt{4\times 6+1}-\sqrt{6-2}=3$$

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

$$3=3$$

Whakakapia te 2 mō te y i te whārite \sqrt{4y+1}-\sqrt{y-2}=3.

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

Whakarūnātia. Ko te uara y=2 kua ngata te whārite.

$$3=3$$

Rārangihia ngā rongoā katoa o \sqrt{4y+1}=\sqrt{y-2}+3.

$$y=6 y=2$$