Whakaoti i te sqrt{x^2+2x+9}-7=2x

$$\sqrt { x ^ { 2 } + 2 x + 9 } - 7 = 2 x$$

Whakaoti mō x

$$x=-2$$

Ngā Upane Otinga

Me tango -7 mai i ngā taha e rua o te whārite.

$$\sqrt{x^{2}+2x+9}=2x+7$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{x^{2}+2x+9}\right)^{2}=\left(2x+7\right)^{2}$$

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

$$x^{2}+2x+9=\left(2x+7\right)^{2}$$

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

$$x^{2}+2x+9=4x^{2}+28x+49$$

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

$$x^{2}+2x+9-4x^{2}=28x+49$$

Pahekotia te x^{2} me -4x^{2}, ka -3x^{2}.

$$-3x^{2}+2x+9=28x+49$$

Tangohia te 28x mai i ngā taha e rua.

$$-3x^{2}+2x+9-28x=49$$

Pahekotia te 2x me -28x, ka -26x.

$$-3x^{2}-26x+9=49$$

Tangohia te 49 mai i ngā taha e rua.

$$-3x^{2}-26x+9-49=0$$

Tangohia te 49 i te 9, ka -40.

$$-3x^{2}-26x-40=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -3x^{2}+ax+bx-40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-26 ab=-3\left(-40\right)=120$$

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 120.

$$-1,-120 -2,-60 -3,-40 -4,-30 -5,-24 -6,-20 -8,-15 -10,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-120=-121 -2-60=-62 -3-40=-43 -4-30=-34 -5-24=-29 -6-20=-26 -8-15=-23 -10-12=-22$$

Ko te otinga te takirua ka hoatu i te tapeke -26.

$$a=-6 b=-20$$

Tuhia anō te -3x^{2}-26x-40 hei \left(-3x^{2}-6x\right)+\left(-20x-40\right).

$$\left(-3x^{2}-6x\right)+\left(-20x-40\right)$$

Tauwehea te 3x i te tuatahi me te 20 i te rōpū tuarua.

$$3x\left(-x-2\right)+20\left(-x-2\right)$$

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

$$\left(-x-2\right)\left(3x+20\right)$$

Hei kimi otinga whārite, me whakaoti te -x-2=0 me te 3x+20=0.

$$x=-2 x=-\frac{20}{3}$$

Whakakapia te -2 mō te x i te whārite \sqrt{x^{2}+2x+9}-7=2x.

$$\sqrt{\left(-2\right)^{2}+2\left(-2\right)+9}-7=2\left(-2\right)$$

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

$$-4=-4$$

Whakakapia te -\frac{20}{3} mō te x i te whārite \sqrt{x^{2}+2x+9}-7=2x.

$$\sqrt{\left(-\frac{20}{3}\right)^{2}+2\left(-\frac{20}{3}\right)+9}-7=2\left(-\frac{20}{3}\right)$$

Whakarūnātia. Ko te uara x=-\frac{20}{3} kāore e ngata ana ki te whārite.

$$-\frac{2}{3}=-\frac{40}{3}$$

Ko te whārite \sqrt{x^{2}+2x+9}=2x+7 he rongoā ahurei.

$$x=-2$$