Whakaoti i te sqrt{5x+16}+x=10

$$\sqrt{ 5x+16 } +x=10$$

Whakaoti mō x

$$x=4$$

Ngā Upane Otinga

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

$$\sqrt{5x+16}=10-x$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{5x+16}\right)^{2}=\left(10-x\right)^{2}$$

Tātaihia te \sqrt{5x+16} mā te pū o 2, kia riro ko 5x+16.

$$5x+16=\left(10-x\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(10-x\right)^{2}.

$$5x+16=100-20x+x^{2}$$

Tangohia te 100 mai i ngā taha e rua.

$$5x+16-100=-20x+x^{2}$$

Tangohia te 100 i te 16, ka -84.

$$5x-84=-20x+x^{2}$$

Me tāpiri te 20x ki ngā taha e rua.

$$5x-84+20x=x^{2}$$

Pahekotia te 5x me 20x, ka 25x.

$$25x-84=x^{2}$$

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

$$25x-84-x^{2}=0$$

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

$$-x^{2}+25x-84=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-84. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=25 ab=-\left(-84\right)=84$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Tātaihia te tapeke mō ia takirua.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

Ko te otinga te takirua ka hoatu i te tapeke 25.

$$a=21 b=4$$

Tuhia anō te -x^{2}+25x-84 hei \left(-x^{2}+21x\right)+\left(4x-84\right).

$$\left(-x^{2}+21x\right)+\left(4x-84\right)$$

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

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

Whakatauwehea atu te kīanga pātahi x-21 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$x=21 x=4$$

Whakakapia te 21 mō te x i te whārite \sqrt{5x+16}+x=10.

$$\sqrt{5\times 21+16}+21=10$$

Whakarūnātia. Ko te uara x=21 kāore e ngata ana ki te whārite.

$$32=10$$

Whakakapia te 4 mō te x i te whārite \sqrt{5x+16}+x=10.

$$\sqrt{5\times 4+16}+4=10$$

Whakarūnātia. Ko te uara x=4 kua ngata te whārite.

$$10=10$$

Ko te whārite \sqrt{5x+16}=10-x he rongoā ahurei.

$$x=4$$