Whakaoti i te x/4+8/x=3

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

Whakaoti mō x

$$x=4$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4x, arā, te tauraro pātahi he tino iti rawa te kitea o 4,x.

$$xx+4\times 8=12x$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}+4\times 8=12x$$

Whakareatia te 4 ki te 8, ka 32.

$$x^{2}+32=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+32-12x=0$$

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

$$x^{2}-12x+32=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-12x+32 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-12 ab=32$$

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 32.

$$-1,-32 -2,-16 -4,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-32=-33 -2-16=-18 -4-8=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-8 b=-4$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

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

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

$$x=8 x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4x, arā, te tauraro pātahi he tino iti rawa te kitea o 4,x.

$$xx+4\times 8=12x$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}+4\times 8=12x$$

Whakareatia te 4 ki te 8, ka 32.

$$x^{2}+32=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+32-12x=0$$

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

$$x^{2}-12x+32=0$$

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

$$a+b=-12 ab=1\times 32=32$$

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 32.

$$-1,-32 -2,-16 -4,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-32=-33 -2-16=-18 -4-8=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-8 b=-4$$

Tuhia anō te x^{2}-12x+32 hei \left(x^{2}-8x\right)+\left(-4x+32\right).

$$\left(x^{2}-8x\right)+\left(-4x+32\right)$$

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

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

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$x=8 x=4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4x, arā, te tauraro pātahi he tino iti rawa te kitea o 4,x.

$$xx+4\times 8=12x$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}+4\times 8=12x$$

Whakareatia te 4 ki te 8, ka 32.

$$x^{2}+32=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+32-12x=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$x^{2}-12x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -12 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 32}}{2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 32}}{2}$$

Whakareatia -4 ki te 32.

$$x=\frac{-\left(-12\right)±\sqrt{144-128}}{2}$$

Tāpiri 144 ki te -128.

$$x=\frac{-\left(-12\right)±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-\left(-12\right)±4}{2}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±4}{2}$$

Nā, me whakaoti te whārite x=\frac{12±4}{2} ina he tāpiri te ±. Tāpiri 12 ki te 4.

$$x=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{12±4}{2} ina he tango te ±. Tango 4 mai i 12.

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=8 x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4x, arā, te tauraro pātahi he tino iti rawa te kitea o 4,x.

$$xx+4\times 8=12x$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}+4\times 8=12x$$

Whakareatia te 4 ki te 8, ka 32.

$$x^{2}+32=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+32-12x=0$$

Tangohia te 32 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}-12x=-32$$

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=-32+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=-32+36$$

Tāpiri -32 ki te 36.

$$x^{2}-12x+36=4$$

Tauwehea x^{2}-12x+36. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=4$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x-6=2 x-6=-2$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=8 x=4$$