Whakaoti i te sqrt{4y+20}-sqrt{y-4}=6

$$\sqrt { 4 y + 20 } - \sqrt { y - 4 } = 6$$

Whakaoti mō y

$$y=20$$

$$y=4$$

Ngā Upane Otinga

Me tango -\sqrt{y-4} mai i ngā taha e rua o te whārite.

$$\sqrt{4y+20}=6+\sqrt{y-4}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{4y+20}\right)^{2}=\left(6+\sqrt{y-4}\right)^{2}$$

Tātaihia te \sqrt{4y+20} mā te pū o 2, kia riro ko 4y+20.

$$4y+20=\left(6+\sqrt{y-4}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(6+\sqrt{y-4}\right)^{2}.

$$4y+20=36+12\sqrt{y-4}+\left(\sqrt{y-4}\right)^{2}$$

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

$$4y+20=36+12\sqrt{y-4}+y-4$$

Tangohia te 4 i te 36, ka 32.

$$4y+20=32+12\sqrt{y-4}+y$$

Me tango 32+y mai i ngā taha e rua o te whārite.

$$4y+20-\left(32+y\right)=12\sqrt{y-4}$$

Hei kimi i te tauaro o 32+y, kimihia te tauaro o ia taurangi.

$$4y+20-32-y=12\sqrt{y-4}$$

Tangohia te 32 i te 20, ka -12.

$$4y-12-y=12\sqrt{y-4}$$

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

$$3y-12=12\sqrt{y-4}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(3y-12\right)^{2}=\left(12\sqrt{y-4}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3y-12\right)^{2}.

$$9y^{2}-72y+144=\left(12\sqrt{y-4}\right)^{2}$$

Whakarohaina te \left(12\sqrt{y-4}\right)^{2}.

$$9y^{2}-72y+144=12^{2}\left(\sqrt{y-4}\right)^{2}$$

Tātaihia te 12 mā te pū o 2, kia riro ko 144.

$$9y^{2}-72y+144=144\left(\sqrt{y-4}\right)^{2}$$

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

$$9y^{2}-72y+144=144\left(y-4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 144 ki te y-4.

$$9y^{2}-72y+144=144y-576$$

Tangohia te 144y mai i ngā taha e rua.

$$9y^{2}-72y+144-144y=-576$$

Pahekotia te -72y me -144y, ka -216y.

$$9y^{2}-216y+144=-576$$

Me tāpiri te 576 ki ngā taha e rua.

$$9y^{2}-216y+144+576=0$$

Tāpirihia te 144 ki te 576, ka 720.

$$9y^{2}-216y+720=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -216 mō b, me 720 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-216\right)±\sqrt{\left(-216\right)^{2}-4\times 9\times 720}}{2\times 9}$$

Pūrua -216.

$$y=\frac{-\left(-216\right)±\sqrt{46656-4\times 9\times 720}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$y=\frac{-\left(-216\right)±\sqrt{46656-36\times 720}}{2\times 9}$$

Whakareatia -36 ki te 720.

$$y=\frac{-\left(-216\right)±\sqrt{46656-25920}}{2\times 9}$$

Tāpiri 46656 ki te -25920.

$$y=\frac{-\left(-216\right)±\sqrt{20736}}{2\times 9}$$

Tuhia te pūtakerua o te 20736.

$$y=\frac{-\left(-216\right)±144}{2\times 9}$$

Ko te tauaro o -216 ko 216.

$$y=\frac{216±144}{2\times 9}$$

Whakareatia 2 ki te 9.

$$y=\frac{216±144}{18}$$

Nā, me whakaoti te whārite y=\frac{216±144}{18} ina he tāpiri te ±. Tāpiri 216 ki te 144.

$$y=\frac{360}{18}$$

Whakawehe 360 ki te 18.

$$y=20$$

Nā, me whakaoti te whārite y=\frac{216±144}{18} ina he tango te ±. Tango 144 mai i 216.

$$y=\frac{72}{18}$$

Whakawehe 72 ki te 18.

$$y=4$$

Kua oti te whārite te whakatau.

$$y=20 y=4$$

Whakakapia te 20 mō te y i te whārite \sqrt{4y+20}-\sqrt{y-4}=6.

$$\sqrt{4\times 20+20}-\sqrt{20-4}=6$$

Whakarūnātia. Ko te uara y=20 kua ngata te whārite.

$$6=6$$

Whakakapia te 4 mō te y i te whārite \sqrt{4y+20}-\sqrt{y-4}=6.

$$\sqrt{4\times 4+20}-\sqrt{4-4}=6$$

Whakarūnātia. Ko te uara y=4 kua ngata te whārite.

$$6=6$$

Rārangihia ngā rongoā katoa o \sqrt{4y+20}=\sqrt{y-4}+6.

$$y=20 y=4$$