Whakaoti i te 3sqrt{2x-3}+2sqrt{7-x}=11

$$3 \sqrt{ 2x-3 } +2 \sqrt{ 7-x } = 11$$

Whakaoti mō x

$$x=6$$

Ngā Upane Otinga

Me tango 2\sqrt{7-x} mai i ngā taha e rua o te whārite.

$$3\sqrt{2x-3}=11-2\sqrt{7-x}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(3\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

Whakarohaina te \left(3\sqrt{2x-3}\right)^{2}.

$$3^{2}\left(\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$9\left(\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

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

$$9\left(2x-3\right)=\left(11-2\sqrt{7-x}\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9 ki te 2x-3.

$$18x-27=\left(11-2\sqrt{7-x}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(11-2\sqrt{7-x}\right)^{2}.

$$18x-27=121-44\sqrt{7-x}+4\left(\sqrt{7-x}\right)^{2}$$

Tātaihia te \sqrt{7-x} mā te pū o 2, kia riro ko 7-x.

$$18x-27=121-44\sqrt{7-x}+4\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 7-x.

$$18x-27=121-44\sqrt{7-x}+28-4x$$

Tāpirihia te 121 ki te 28, ka 149.

$$18x-27=149-44\sqrt{7-x}-4x$$

Me tango 149-4x mai i ngā taha e rua o te whārite.

$$18x-27-\left(149-4x\right)=-44\sqrt{7-x}$$

Hei kimi i te tauaro o 149-4x, kimihia te tauaro o ia taurangi.

$$18x-27-149+4x=-44\sqrt{7-x}$$

Tangohia te 149 i te -27, ka -176.

$$18x-176+4x=-44\sqrt{7-x}$$

Pahekotia te 18x me 4x, ka 22x.

$$22x-176=-44\sqrt{7-x}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(22x-176\right)^{2}=\left(-44\sqrt{7-x}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(22x-176\right)^{2}.

$$484x^{2}-7744x+30976=\left(-44\sqrt{7-x}\right)^{2}$$

Whakarohaina te \left(-44\sqrt{7-x}\right)^{2}.

$$484x^{2}-7744x+30976=\left(-44\right)^{2}\left(\sqrt{7-x}\right)^{2}$$

Tātaihia te -44 mā te pū o 2, kia riro ko 1936.

$$484x^{2}-7744x+30976=1936\left(\sqrt{7-x}\right)^{2}$$

Tātaihia te \sqrt{7-x} mā te pū o 2, kia riro ko 7-x.

$$484x^{2}-7744x+30976=1936\left(7-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1936 ki te 7-x.

$$484x^{2}-7744x+30976=13552-1936x$$

Tangohia te 13552 mai i ngā taha e rua.

$$484x^{2}-7744x+30976-13552=-1936x$$

Tangohia te 13552 i te 30976, ka 17424.

$$484x^{2}-7744x+17424=-1936x$$

Me tāpiri te 1936x ki ngā taha e rua.

$$484x^{2}-7744x+17424+1936x=0$$

Pahekotia te -7744x me 1936x, ka -5808x.

$$484x^{2}-5808x+17424=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 484 mō a, -5808 mō b, me 17424 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5808\right)±\sqrt{\left(-5808\right)^{2}-4\times 484\times 17424}}{2\times 484}$$

Pūrua -5808.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-4\times 484\times 17424}}{2\times 484}$$

Whakareatia -4 ki te 484.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-1936\times 17424}}{2\times 484}$$

Whakareatia -1936 ki te 17424.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-33732864}}{2\times 484}$$

Tāpiri 33732864 ki te -33732864.

$$x=\frac{-\left(-5808\right)±\sqrt{0}}{2\times 484}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-5808}{2\times 484}$$

Ko te tauaro o -5808 ko 5808.

$$x=\frac{5808}{2\times 484}$$

Whakareatia 2 ki te 484.

$$x=\frac{5808}{968}$$

Whakawehe 5808 ki te 968.

$$x=6$$

Whakakapia te 6 mō te x i te whārite 3\sqrt{2x-3}+2\sqrt{7-x}=11.

$$3\sqrt{2\times 6-3}+2\sqrt{7-6}=11$$

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

$$11=11$$

Ko te whārite 3\sqrt{2x-3}=-2\sqrt{7-x}+11 he rongoā ahurei.

$$x=6$$