Whakaoti i te sqrt{2x+9}-sqrt{x-4}-3=0

$$\sqrt { 2 x + 9 } - \sqrt { x - 4 } - 3 = 0$$

Whakaoti mō x

$$x=20$$

$$x=8$$

Ngā Upane Otinga

Me tango -\sqrt{x-4}-3 mai i ngā taha e rua o te whārite.

$$\sqrt{2x+9}=-\left(-\sqrt{x-4}-3\right)$$

Hei kimi i te tauaro o -\sqrt{x-4}-3, kimihia te tauaro o ia taurangi.

$$\sqrt{2x+9}=-\left(-\sqrt{x-4}\right)-\left(-3\right)$$

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

$$\sqrt{2x+9}=\sqrt{x-4}-\left(-3\right)$$

Ko te tauaro o -3 ko 3.

$$\sqrt{2x+9}=\sqrt{x-4}+3$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{2x+9}\right)^{2}=\left(\sqrt{x-4}+3\right)^{2}$$

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

$$2x+9=\left(\sqrt{x-4}+3\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(\sqrt{x-4}+3\right)^{2}.

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

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

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

Tāpirihia te -4 ki te 9, ka 5.

$$2x+9=x+5+6\sqrt{x-4}$$

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

$$2x+9-\left(x+5\right)=6\sqrt{x-4}$$

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

$$2x+9-x-5=6\sqrt{x-4}$$

Pahekotia te 2x me -x, ka x.

$$x+9-5=6\sqrt{x-4}$$

Tangohia te 5 i te 9, ka 4.

$$x+4=6\sqrt{x-4}$$

Pūruatia ngā taha e rua o te whārite.

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

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

$$x^{2}+8x+16=\left(6\sqrt{x-4}\right)^{2}$$

Whakarohaina te \left(6\sqrt{x-4}\right)^{2}.

$$x^{2}+8x+16=6^{2}\left(\sqrt{x-4}\right)^{2}$$

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

$$x^{2}+8x+16=36\left(\sqrt{x-4}\right)^{2}$$

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

$$x^{2}+8x+16=36\left(x-4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 36 ki te x-4.

$$x^{2}+8x+16=36x-144$$

Tangohia te 36x mai i ngā taha e rua.

$$x^{2}+8x+16-36x=-144$$

Pahekotia te 8x me -36x, ka -28x.

$$x^{2}-28x+16=-144$$

Me tāpiri te 144 ki ngā taha e rua.

$$x^{2}-28x+16+144=0$$

Tāpirihia te 16 ki te 144, ka 160.

$$x^{2}-28x+160=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-28x+160 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-28 ab=160$$

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 160.

$$-1,-160 -2,-80 -4,-40 -5,-32 -8,-20 -10,-16$$

Tātaihia te tapeke mō ia takirua.

$$-1-160=-161 -2-80=-82 -4-40=-44 -5-32=-37 -8-20=-28 -10-16=-26$$

Ko te otinga te takirua ka hoatu i te tapeke -28.

$$a=-20 b=-8$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-20\right)\left(x-8\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x-8=0.

$$x=20 x=8$$

Whakakapia te 20 mō te x i te whārite \sqrt{2x+9}-\sqrt{x-4}-3=0.

$$\sqrt{2\times 20+9}-\sqrt{20-4}-3=0$$

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

$$0=0$$

Whakakapia te 8 mō te x i te whārite \sqrt{2x+9}-\sqrt{x-4}-3=0.

$$\sqrt{2\times 8+9}-\sqrt{8-4}-3=0$$

Whakarūnātia. Ko te uara x=8 kua ngata te whārite.

$$0=0$$

Rārangihia ngā rongoā katoa o \sqrt{2x+9}=\sqrt{x-4}+3.

$$x=20 x=8$$