Whakaoti i te sqrt{333+49x}-x=19

$$\sqrt { 333 + 49 x } - x = 19$$

Whakaoti mō x

$$x=7$$

$$x=4$$

Ngā Upane Otinga

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

$$\sqrt{333+49x}=19+x$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{333+49x}\right)^{2}=\left(19+x\right)^{2}$$

Tātaihia te \sqrt{333+49x} mā te pū o 2, kia riro ko 333+49x.

$$333+49x=\left(19+x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(19+x\right)^{2}.

$$333+49x=361+38x+x^{2}$$

Tangohia te 361 mai i ngā taha e rua.

$$333+49x-361=38x+x^{2}$$

Tangohia te 361 i te 333, ka -28.

$$-28+49x=38x+x^{2}$$

Tangohia te 38x mai i ngā taha e rua.

$$-28+49x-38x=x^{2}$$

Pahekotia te 49x me -38x, ka 11x.

$$-28+11x=x^{2}$$

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

$$-28+11x-x^{2}=0$$

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

$$-x^{2}+11x-28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=-\left(-28\right)=28$$

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 28.

$$1,28 2,14 4,7$$

Tātaihia te tapeke mō ia takirua.

$$1+28=29 2+14=16 4+7=11$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=7 b=4$$

Tuhia anō te -x^{2}+11x-28 hei \left(-x^{2}+7x\right)+\left(4x-28\right).

$$\left(-x^{2}+7x\right)+\left(4x-28\right)$$

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

$$-x\left(x-7\right)+4\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(-x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te -x+4=0.

$$x=7 x=4$$

Whakakapia te 7 mō te x i te whārite \sqrt{333+49x}-x=19.

$$\sqrt{333+49\times 7}-7=19$$

Whakarūnātia. Ko te uara x=7 kua ngata te whārite.

$$19=19$$

Whakakapia te 4 mō te x i te whārite \sqrt{333+49x}-x=19.

$$\sqrt{333+49\times 4}-4=19$$

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

$$19=19$$

Rārangihia ngā rongoā katoa o \sqrt{49x+333}=x+19.

$$x=7 x=4$$