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

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

Whakaoti mō x

$$x=\frac{3}{4}=0.75$$

Ngā Upane Otinga

Me tango 1 mai i ngā taha e rua o te whārite.

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

Pūruatia ngā taha e rua o te whārite.

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

Whakarohaina te \left(4\sqrt{1-x}\right)^{2}.

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

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$16\left(\sqrt{1-x}\right)^{2}=\left(4x-1\right)^{2}$$

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

$$16\left(1-x\right)=\left(4x-1\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te 1-x.

$$16-16x=\left(4x-1\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(4x-1\right)^{2}.

$$16-16x=16x^{2}-8x+1$$

Tangohia te 16x^{2} mai i ngā taha e rua.

$$16-16x-16x^{2}=-8x+1$$

Me tāpiri te 8x ki ngā taha e rua.

$$16-16x-16x^{2}+8x=1$$

Pahekotia te -16x me 8x, ka -8x.

$$16-8x-16x^{2}=1$$

Tangohia te 1 mai i ngā taha e rua.

$$16-8x-16x^{2}-1=0$$

Tangohia te 1 i te 16, ka 15.

$$15-8x-16x^{2}=0$$

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

$$-16x^{2}-8x+15=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -16x^{2}+ax+bx+15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=-16\times 15=-240$$

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 -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Tātaihia te tapeke mō ia takirua.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=12 b=-20$$

Tuhia anō te -16x^{2}-8x+15 hei \left(-16x^{2}+12x\right)+\left(-20x+15\right).

$$\left(-16x^{2}+12x\right)+\left(-20x+15\right)$$

Tauwehea te 4x i te tuatahi me te 5 i te rōpū tuarua.

$$4x\left(-4x+3\right)+5\left(-4x+3\right)$$

Whakatauwehea atu te kīanga pātahi -4x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-4x+3\right)\left(4x+5\right)$$

Hei kimi otinga whārite, me whakaoti te -4x+3=0 me te 4x+5=0.

$$x=\frac{3}{4} x=-\frac{5}{4}$$

Whakakapia te \frac{3}{4} mō te x i te whārite 1+4\sqrt{1-x}=4x.

$$1+4\sqrt{1-\frac{3}{4}}=4\times \frac{3}{4}$$

Whakarūnātia. Ko te uara x=\frac{3}{4} kua ngata te whārite.

$$3=3$$

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

$$1+4\sqrt{1-\left(-\frac{5}{4}\right)}=4\left(-\frac{5}{4}\right)$$

Whakarūnātia. Ko te uara x=-\frac{5}{4} kāore e ngata ana ki te whārite nā te mea e rerekē ngā tohu o te taha maui me te taha katau.

$$7=-5$$

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

$$x=\frac{3}{4}$$