Whakaoti i te x+sqrt{4x+1}=5

$$x + \sqrt { 4 x + 1 } = 5$$

Whakaoti mō x

$$x=2$$

Ngā Upane Otinga

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

$$\sqrt{4x+1}=5-x$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{4x+1}\right)^{2}=\left(5-x\right)^{2}$$

Tātaihia te \sqrt{4x+1} mā te pū o 2, kia riro ko 4x+1.

$$4x+1=\left(5-x\right)^{2}$$

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

$$4x+1=25-10x+x^{2}$$

Tangohia te 25 mai i ngā taha e rua.

$$4x+1-25=-10x+x^{2}$$

Tangohia te 25 i te 1, ka -24.

$$4x-24=-10x+x^{2}$$

Me tāpiri te 10x ki ngā taha e rua.

$$4x-24+10x=x^{2}$$

Pahekotia te 4x me 10x, ka 14x.

$$14x-24=x^{2}$$

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

$$14x-24-x^{2}=0$$

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

$$-x^{2}+14x-24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=-\left(-24\right)=24$$

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 24.

$$1,24 2,12 3,8 4,6$$

Tātaihia te tapeke mō ia takirua.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=12 b=2$$

Tuhia anō te -x^{2}+14x-24 hei \left(-x^{2}+12x\right)+\left(2x-24\right).

$$\left(-x^{2}+12x\right)+\left(2x-24\right)$$

Tauwehea te -x i te tuatahi me te 2 i te rōpū tuarua.

$$-x\left(x-12\right)+2\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(-x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te -x+2=0.

$$x=12 x=2$$

Whakakapia te 12 mō te x i te whārite x+\sqrt{4x+1}=5.

$$12+\sqrt{4\times 12+1}=5$$

Whakarūnātia. Ko te uara x=12 kāore e ngata ana ki te whārite.

$$19=5$$

Whakakapia te 2 mō te x i te whārite x+\sqrt{4x+1}=5.

$$2+\sqrt{4\times 2+1}=5$$

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

$$5=5$$

Ko te whārite \sqrt{4x+1}=5-x he rongoā ahurei.

$$x=2$$