Whakaoti i te sqrt{4y+12}-sqrt{y-6}=6

$$\sqrt { 4 y + 12 } - \sqrt { y - 6 } = 6$$

Whakaoti mō y

$$y=22$$

$$y=6$$

Ngā Upane Otinga

Me tango -\sqrt{y-6} mai i ngā taha e rua o te whārite.

$$\sqrt{4y+12}=6+\sqrt{y-6}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{4y+12}\right)^{2}=\left(6+\sqrt{y-6}\right)^{2}$$

Tātaihia te \sqrt{4y+12} mā te pū o 2, kia riro ko 4y+12.

$$4y+12=\left(6+\sqrt{y-6}\right)^{2}$$

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

$$4y+12=36+12\sqrt{y-6}+\left(\sqrt{y-6}\right)^{2}$$

Tātaihia te \sqrt{y-6} mā te pū o 2, kia riro ko y-6.

$$4y+12=36+12\sqrt{y-6}+y-6$$

Tangohia te 6 i te 36, ka 30.

$$4y+12=30+12\sqrt{y-6}+y$$

Me tango 30+y mai i ngā taha e rua o te whārite.

$$4y+12-\left(30+y\right)=12\sqrt{y-6}$$

Hei kimi i te tauaro o 30+y, kimihia te tauaro o ia taurangi.

$$4y+12-30-y=12\sqrt{y-6}$$

Tangohia te 30 i te 12, ka -18.

$$4y-18-y=12\sqrt{y-6}$$

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

$$3y-18=12\sqrt{y-6}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(3y-18\right)^{2}=\left(12\sqrt{y-6}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(3y-18\right)^{2}.

$$9y^{2}-108y+324=\left(12\sqrt{y-6}\right)^{2}$$

Whakarohaina te \left(12\sqrt{y-6}\right)^{2}.

$$9y^{2}-108y+324=12^{2}\left(\sqrt{y-6}\right)^{2}$$

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

$$9y^{2}-108y+324=144\left(\sqrt{y-6}\right)^{2}$$

Tātaihia te \sqrt{y-6} mā te pū o 2, kia riro ko y-6.

$$9y^{2}-108y+324=144\left(y-6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 144 ki te y-6.

$$9y^{2}-108y+324=144y-864$$

Tangohia te 144y mai i ngā taha e rua.

$$9y^{2}-108y+324-144y=-864$$

Pahekotia te -108y me -144y, ka -252y.

$$9y^{2}-252y+324=-864$$

Me tāpiri te 864 ki ngā taha e rua.

$$9y^{2}-252y+324+864=0$$

Tāpirihia te 324 ki te 864, ka 1188.

$$9y^{2}-252y+1188=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -252 mō b, me 1188 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-252\right)±\sqrt{\left(-252\right)^{2}-4\times 9\times 1188}}{2\times 9}$$

Pūrua -252.

$$y=\frac{-\left(-252\right)±\sqrt{63504-4\times 9\times 1188}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$y=\frac{-\left(-252\right)±\sqrt{63504-36\times 1188}}{2\times 9}$$

Whakareatia -36 ki te 1188.

$$y=\frac{-\left(-252\right)±\sqrt{63504-42768}}{2\times 9}$$

Tāpiri 63504 ki te -42768.

$$y=\frac{-\left(-252\right)±\sqrt{20736}}{2\times 9}$$

Tuhia te pūtakerua o te 20736.

$$y=\frac{-\left(-252\right)±144}{2\times 9}$$

Ko te tauaro o -252 ko 252.

$$y=\frac{252±144}{2\times 9}$$

Whakareatia 2 ki te 9.

$$y=\frac{252±144}{18}$$

Nā, me whakaoti te whārite y=\frac{252±144}{18} ina he tāpiri te ±. Tāpiri 252 ki te 144.

$$y=\frac{396}{18}$$

Whakawehe 396 ki te 18.

$$y=22$$

Nā, me whakaoti te whārite y=\frac{252±144}{18} ina he tango te ±. Tango 144 mai i 252.

$$y=\frac{108}{18}$$

Whakawehe 108 ki te 18.

$$y=6$$

Kua oti te whārite te whakatau.

$$y=22 y=6$$

Whakakapia te 22 mō te y i te whārite \sqrt{4y+12}-\sqrt{y-6}=6.

$$\sqrt{4\times 22+12}-\sqrt{22-6}=6$$

Whakarūnātia. Ko te uara y=22 kua ngata te whārite.

$$6=6$$

Whakakapia te 6 mō te y i te whārite \sqrt{4y+12}-\sqrt{y-6}=6.

$$\sqrt{4\times 6+12}-\sqrt{6-6}=6$$

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

$$6=6$$

Rārangihia ngā rongoā katoa o \sqrt{4y+12}=\sqrt{y-6}+6.

$$y=22 y=6$$