Whakaoti i te sqrt{7x+67}=2x+5

$$\sqrt { 7 x + 67 } = 2 x + 5$$

Whakaoti mō x

$$x=2$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{7x+67}\right)^{2}=\left(2x+5\right)^{2}$$

Tātaihia te \sqrt{7x+67} mā te pū o 2, kia riro ko 7x+67.

$$7x+67=\left(2x+5\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+5\right)^{2}.

$$7x+67=4x^{2}+20x+25$$

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

$$7x+67-4x^{2}=20x+25$$

Tangohia te 20x mai i ngā taha e rua.

$$7x+67-4x^{2}-20x=25$$

Pahekotia te 7x me -20x, ka -13x.

$$-13x+67-4x^{2}=25$$

Tangohia te 25 mai i ngā taha e rua.

$$-13x+67-4x^{2}-25=0$$

Tangohia te 25 i te 67, ka 42.

$$-13x+42-4x^{2}=0$$

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

$$-4x^{2}-13x+42=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -4x^{2}+ax+bx+42. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-13 ab=-4\times 42=-168$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Tātaihia te tapeke mō ia takirua.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -13.

$$a=8 b=-21$$

Tuhia anō te -4x^{2}-13x+42 hei \left(-4x^{2}+8x\right)+\left(-21x+42\right).

$$\left(-4x^{2}+8x\right)+\left(-21x+42\right)$$

Tauwehea te 4x i te tuatahi me te 21 i te rōpū tuarua.

$$4x\left(-x+2\right)+21\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(4x+21\right)$$

Hei kimi otinga whārite, me whakaoti te -x+2=0 me te 4x+21=0.

$$x=2 x=-\frac{21}{4}$$

Whakakapia te 2 mō te x i te whārite \sqrt{7x+67}=2x+5.

$$\sqrt{7\times 2+67}=2\times 2+5$$

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

$$9=9$$

Whakakapia te -\frac{21}{4} mō te x i te whārite \sqrt{7x+67}=2x+5.

$$\sqrt{7\left(-\frac{21}{4}\right)+67}=2\left(-\frac{21}{4}\right)+5$$

Whakarūnātia. Ko te uara x=-\frac{21}{4} kāore e ngata ana ki te whārite nā te mea e rerekē ngā tohu o te taha maui me te taha katau.

$$\frac{11}{2}=-\frac{11}{2}$$

Ko te whārite \sqrt{7x+67}=2x+5 he rongoā ahurei.

$$x=2$$