Whakaoti i te sqrt{3x+12}-1=sqrt{5x+9}

$$\sqrt{ 3x+12 } -1= \sqrt{ 5x+9 }$$

Whakaoti mō x

$$x=-1$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{3x+12}-1\right)^{2}=\left(\sqrt{5x+9}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{3x+12}-1\right)^{2}.

$$\left(\sqrt{3x+12}\right)^{2}-2\sqrt{3x+12}+1=\left(\sqrt{5x+9}\right)^{2}$$

Tātaihia te \sqrt{3x+12} mā te pū o 2, kia riro ko 3x+12.

$$3x+12-2\sqrt{3x+12}+1=\left(\sqrt{5x+9}\right)^{2}$$

Tāpirihia te 12 ki te 1, ka 13.

$$3x+13-2\sqrt{3x+12}=\left(\sqrt{5x+9}\right)^{2}$$

Tātaihia te \sqrt{5x+9} mā te pū o 2, kia riro ko 5x+9.

$$3x+13-2\sqrt{3x+12}=5x+9$$

Me tango 3x+13 mai i ngā taha e rua o te whārite.

$$-2\sqrt{3x+12}=5x+9-\left(3x+13\right)$$

Hei kimi i te tauaro o 3x+13, kimihia te tauaro o ia taurangi.

$$-2\sqrt{3x+12}=5x+9-3x-13$$

Pahekotia te 5x me -3x, ka 2x.

$$-2\sqrt{3x+12}=2x+9-13$$

Tangohia te 13 i te 9, ka -4.

$$-2\sqrt{3x+12}=2x-4$$

Pūruatia ngā taha e rua o te whārite.

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

Whakarohaina te \left(-2\sqrt{3x+12}\right)^{2}.

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

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

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

Tātaihia te \sqrt{3x+12} mā te pū o 2, kia riro ko 3x+12.

$$4\left(3x+12\right)=\left(2x-4\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 3x+12.

$$12x+48=\left(2x-4\right)^{2}$$

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

$$12x+48=4x^{2}-16x+16$$

Tangohia te 4x^{2} mai i ngā taha e rua.

$$12x+48-4x^{2}=-16x+16$$

Me tāpiri te 16x ki ngā taha e rua.

$$12x+48-4x^{2}+16x=16$$

Pahekotia te 12x me 16x, ka 28x.

$$28x+48-4x^{2}=16$$

Tangohia te 16 mai i ngā taha e rua.

$$28x+48-4x^{2}-16=0$$

Tangohia te 16 i te 48, ka 32.

$$28x+32-4x^{2}=0$$

Whakawehea ngā taha e rua ki te 4.

$$7x+8-x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+7x+8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=-8=-8$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=8 b=-1$$

Tuhia anō te -x^{2}+7x+8 hei \left(-x^{2}+8x\right)+\left(-x+8\right).

$$\left(-x^{2}+8x\right)+\left(-x+8\right)$$

Tauwehea te -x i te tuatahi me te -1 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-8\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te -x-1=0.

$$x=8 x=-1$$

Whakakapia te 8 mō te x i te whārite \sqrt{3x+12}-1=\sqrt{5x+9}.

$$\sqrt{3\times 8+12}-1=\sqrt{5\times 8+9}$$

Whakarūnātia. Ko te uara x=8 kāore e ngata ana ki te whārite.

$$5=7$$

Whakakapia te -1 mō te x i te whārite \sqrt{3x+12}-1=\sqrt{5x+9}.

$$\sqrt{3\left(-1\right)+12}-1=\sqrt{5\left(-1\right)+9}$$

Whakarūnātia. Ko te uara x=-1 kua ngata te whārite.

$$2=2$$

Ko te whārite \sqrt{3x+12}-1=\sqrt{5x+9} he rongoā ahurei.

$$x=-1$$