Whakaoti i te sqrt{2x+13}-3x=9

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

Whakaoti mō x

$$x=-2$$

Ngā Upane Otinga

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

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

Pūruatia ngā taha e rua o te whārite.

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

Tātaihia te \sqrt{2x+13} mā te pū o 2, kia riro ko 2x+13.

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(9+3x\right)^{2}.

$$2x+13=81+54x+9x^{2}$$

Tangohia te 81 mai i ngā taha e rua.

$$2x+13-81=54x+9x^{2}$$

Tangohia te 81 i te 13, ka -68.

$$2x-68=54x+9x^{2}$$

Tangohia te 54x mai i ngā taha e rua.

$$2x-68-54x=9x^{2}$$

Pahekotia te 2x me -54x, ka -52x.

$$-52x-68=9x^{2}$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$-52x-68-9x^{2}=0$$

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

$$-9x^{2}-52x-68=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -9x^{2}+ax+bx-68. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-52 ab=-9\left(-68\right)=612$$

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 612.

$$-1,-612 -2,-306 -3,-204 -4,-153 -6,-102 -9,-68 -12,-51 -17,-36 -18,-34$$

Tātaihia te tapeke mō ia takirua.

$$-1-612=-613 -2-306=-308 -3-204=-207 -4-153=-157 -6-102=-108 -9-68=-77 -12-51=-63 -17-36=-53 -18-34=-52$$

Ko te otinga te takirua ka hoatu i te tapeke -52.

$$a=-18 b=-34$$

Tuhia anō te -9x^{2}-52x-68 hei \left(-9x^{2}-18x\right)+\left(-34x-68\right).

$$\left(-9x^{2}-18x\right)+\left(-34x-68\right)$$

Tauwehea te 9x i te tuatahi me te 34 i te rōpū tuarua.

$$9x\left(-x-2\right)+34\left(-x-2\right)$$

Whakatauwehea atu te kīanga pātahi -x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x-2\right)\left(9x+34\right)$$

Hei kimi otinga whārite, me whakaoti te -x-2=0 me te 9x+34=0.

$$x=-2 x=-\frac{34}{9}$$

Whakakapia te -2 mō te x i te whārite \sqrt{2x+13}-3x=9.

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

Whakarūnātia. Ko te uara x=-2 kua ngata te whārite.

$$9=9$$

Whakakapia te -\frac{34}{9} mō te x i te whārite \sqrt{2x+13}-3x=9.

$$\sqrt{2\left(-\frac{34}{9}\right)+13}-3\left(-\frac{34}{9}\right)=9$$

Whakarūnātia. Ko te uara x=-\frac{34}{9} kāore e ngata ana ki te whārite.

$$\frac{41}{3}=9$$

Ko te whārite \sqrt{2x+13}=3x+9 he rongoā ahurei.

$$x=-2$$