Whakaoti i te sqrt{5n+9}=n-1

$$\sqrt { 5 n + 9 } = n - 1$$

Whakaoti mō n

$$n=8$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

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

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

$$5n+9=\left(n-1\right)^{2}$$

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

$$5n+9=n^{2}-2n+1$$

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

$$5n+9-n^{2}=-2n+1$$

Me tāpiri te 2n ki ngā taha e rua.

$$5n+9-n^{2}+2n=1$$

Pahekotia te 5n me 2n, ka 7n.

$$7n+9-n^{2}=1$$

Tangohia te 1 mai i ngā taha e rua.

$$7n+9-n^{2}-1=0$$

Tangohia te 1 i te 9, ka 8.

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

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

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -n^{2}+an+bn+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 -n^{2}+7n+8 hei \left(-n^{2}+8n\right)+\left(-n+8\right).

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

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

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

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

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

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

$$n=8 n=-1$$

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

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

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

$$7=7$$

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

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

Whakarūnātia. Ko te uara n=-1 kāore e ngata ana ki te whārite nā te mea e rerekē ngā tohu o te taha maui me te taha katau.

$$2=-2$$

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

$$n=8$$