Whakaoti i te sqrt{x-4}-sqrt{4x-27}+sqrt{x-9}=0

$$\sqrt{ x-4 } - \sqrt{ 4x-27 } + \sqrt{ x-9 } =0$$

Whakaoti mō x

$$x=13$$

Ngā Upane Otinga

Me tango -\sqrt{4x-27}+\sqrt{x-9} mai i ngā taha e rua o te whārite.

$$\sqrt{x-4}=-\left(-\sqrt{4x-27}+\sqrt{x-9}\right)$$

Hei kimi i te tauaro o -\sqrt{4x-27}+\sqrt{x-9}, kimihia te tauaro o ia taurangi.

$$\sqrt{x-4}=-\left(-\sqrt{4x-27}\right)-\sqrt{x-9}$$

Ko te tauaro o -\sqrt{4x-27} ko \sqrt{4x-27}.

$$\sqrt{x-4}=\sqrt{4x-27}-\sqrt{x-9}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{x-4}\right)^{2}=\left(\sqrt{4x-27}-\sqrt{x-9}\right)^{2}$$

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

$$x-4=\left(\sqrt{4x-27}-\sqrt{x-9}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{4x-27}-\sqrt{x-9}\right)^{2}.

$$x-4=\left(\sqrt{4x-27}\right)^{2}-2\sqrt{4x-27}\sqrt{x-9}+\left(\sqrt{x-9}\right)^{2}$$

Tātaihia te \sqrt{4x-27} mā te pū o 2, kia riro ko 4x-27.

$$x-4=4x-27-2\sqrt{4x-27}\sqrt{x-9}+\left(\sqrt{x-9}\right)^{2}$$

Tātaihia te \sqrt{x-9} mā te pū o 2, kia riro ko x-9.

$$x-4=4x-27-2\sqrt{4x-27}\sqrt{x-9}+x-9$$

Pahekotia te 4x me x, ka 5x.

$$x-4=5x-27-2\sqrt{4x-27}\sqrt{x-9}-9$$

Tangohia te 9 i te -27, ka -36.

$$x-4=5x-36-2\sqrt{4x-27}\sqrt{x-9}$$

Me tango 5x-36 mai i ngā taha e rua o te whārite.

$$x-4-\left(5x-36\right)=-2\sqrt{4x-27}\sqrt{x-9}$$

Hei kimi i te tauaro o 5x-36, kimihia te tauaro o ia taurangi.

$$x-4-5x+36=-2\sqrt{4x-27}\sqrt{x-9}$$

Pahekotia te x me -5x, ka -4x.

$$-4x-4+36=-2\sqrt{4x-27}\sqrt{x-9}$$

Tāpirihia te -4 ki te 36, ka 32.

$$-4x+32=-2\sqrt{4x-27}\sqrt{x-9}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(-4x+32\right)^{2}=\left(-2\sqrt{4x-27}\sqrt{x-9}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(-4x+32\right)^{2}.

$$16x^{2}-256x+1024=\left(-2\sqrt{4x-27}\sqrt{x-9}\right)^{2}$$

Whakarohaina te \left(-2\sqrt{4x-27}\sqrt{x-9}\right)^{2}.

$$16x^{2}-256x+1024=\left(-2\right)^{2}\left(\sqrt{4x-27}\right)^{2}\left(\sqrt{x-9}\right)^{2}$$

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

$$16x^{2}-256x+1024=4\left(\sqrt{4x-27}\right)^{2}\left(\sqrt{x-9}\right)^{2}$$

Tātaihia te \sqrt{4x-27} mā te pū o 2, kia riro ko 4x-27.

$$16x^{2}-256x+1024=4\left(4x-27\right)\left(\sqrt{x-9}\right)^{2}$$

Tātaihia te \sqrt{x-9} mā te pū o 2, kia riro ko x-9.

$$16x^{2}-256x+1024=4\left(4x-27\right)\left(x-9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 4x-27.

$$16x^{2}-256x+1024=\left(16x-108\right)\left(x-9\right)$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 16x-108 ki ia tau o x-9.

$$16x^{2}-256x+1024=16x^{2}-144x-108x+972$$

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

$$16x^{2}-256x+1024=16x^{2}-252x+972$$

Tangohia te 16x^{2} mai i ngā taha e rua.

$$16x^{2}-256x+1024-16x^{2}=-252x+972$$

Pahekotia te 16x^{2} me -16x^{2}, ka 0.

$$-256x+1024=-252x+972$$

Me tāpiri te 252x ki ngā taha e rua.

$$-256x+1024+252x=972$$

Pahekotia te -256x me 252x, ka -4x.

$$-4x+1024=972$$

Tangohia te 1024 mai i ngā taha e rua.

$$-4x=972-1024$$

Tangohia te 1024 i te 972, ka -52.

$$-4x=-52$$

Whakawehea ngā taha e rua ki te -4.

$$x=\frac{-52}{-4}$$

Whakawehea te -52 ki te -4, kia riro ko 13.

$$x=13$$

Whakakapia te 13 mō te x i te whārite \sqrt{x-4}-\sqrt{4x-27}+\sqrt{x-9}=0.

$$\sqrt{13-4}-\sqrt{4\times 13-27}+\sqrt{13-9}=0$$

Whakarūnātia. Ko te uara x=13 kua ngata te whārite.

$$0=0$$

Ko te whārite \sqrt{x-4}=\sqrt{4x-27}-\sqrt{x-9} he rongoā ahurei.

$$x=13$$