Whakaoti i te 2x-5=sqrt{x^2-7}

$$2 x - 5 = \sqrt { x ^ { 2 } - 7 }$$

Whakaoti mō x

$$x=4$$

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

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

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

$$4x^{2}-20x+25=\left(\sqrt{x^{2}-7}\right)^{2}$$

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

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

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

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

Pahekotia te 4x^{2} me -x^{2}, ka 3x^{2}.

$$3x^{2}-20x+25=-7$$

Me tāpiri te 7 ki ngā taha e rua.

$$3x^{2}-20x+25+7=0$$

Tāpirihia te 25 ki te 7, ka 32.

$$3x^{2}-20x+32=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx+32. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-20 ab=3\times 32=96$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 96.

$$-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20$$

Ko te otinga te takirua ka hoatu i te tapeke -20.

$$a=-12 b=-8$$

Tuhia anō te 3x^{2}-20x+32 hei \left(3x^{2}-12x\right)+\left(-8x+32\right).

$$\left(3x^{2}-12x\right)+\left(-8x+32\right)$$

Tauwehea te 3x i te tuatahi me te -8 i te rōpū tuarua.

$$3x\left(x-4\right)-8\left(x-4\right)$$

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

$$\left(x-4\right)\left(3x-8\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te 3x-8=0.

$$x=4 x=\frac{8}{3}$$

Whakakapia te 4 mō te x i te whārite 2x-5=\sqrt{x^{2}-7}.

$$2\times 4-5=\sqrt{4^{2}-7}$$

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

$$3=3$$

Whakakapia te \frac{8}{3} mō te x i te whārite 2x-5=\sqrt{x^{2}-7}.

$$2\times \frac{8}{3}-5=\sqrt{\left(\frac{8}{3}\right)^{2}-7}$$

Whakarūnātia. Ko te uara x=\frac{8}{3} kua ngata te whārite.

$$\frac{1}{3}=\frac{1}{3}$$

Rārangihia ngā rongoā katoa o 2x-5=\sqrt{x^{2}-7}.

$$x=4 x=\frac{8}{3}$$