Whakaoti i te x^2/8+1=x+7

$$\frac { x ^ { 2 } } { 8 } + 1 = x + 7$$

Whakaoti mō x

$$x=-4$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe

Whakareatia ngā taha e rua o te whārite ki te 8.

$$x^{2}+8=8x+56$$

Tangohia te 8x mai i ngā taha e rua.

$$x^{2}+8-8x=56$$

Tangohia te 56 mai i ngā taha e rua.

$$x^{2}+8-8x-56=0$$

Tangohia te 56 i te 8, ka -48.

$$x^{2}-48-8x=0$$

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

$$x^{2}-8x-48=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-8x-48 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.

$$a+b=-8 ab=-48$$

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 -48.

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

Tātaihia te tapeke mō ia takirua.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

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

$$a=-12 b=4$$

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

$$\left(x-12\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+4=0.

$$x=12 x=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia ngā taha e rua o te whārite ki te 8.

$$x^{2}+8=8x+56$$

Tangohia te 8x mai i ngā taha e rua.

$$x^{2}+8-8x=56$$

Tangohia te 56 mai i ngā taha e rua.

$$x^{2}+8-8x-56=0$$

Tangohia te 56 i te 8, ka -48.

$$x^{2}-48-8x=0$$

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

$$x^{2}-8x-48=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-48. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\left(-48\right)=-48$$

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 -48.

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

Tātaihia te tapeke mō ia takirua.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

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

$$a=-12 b=4$$

Tuhia anō te x^{2}-8x-48 hei \left(x^{2}-12x\right)+\left(4x-48\right).

$$\left(x^{2}-12x\right)+\left(4x-48\right)$$

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

$$x\left(x-12\right)+4\left(x-12\right)$$

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

$$\left(x-12\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+4=0.

$$x=12 x=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 8.

$$x^{2}+8=8x+56$$

Tangohia te 8x mai i ngā taha e rua.

$$x^{2}+8-8x=56$$

Tangohia te 56 mai i ngā taha e rua.

$$x^{2}+8-8x-56=0$$

Tangohia te 56 i te 8, ka -48.

$$x^{2}-48-8x=0$$

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.

$$x^{2}-8x-48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me -48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-48\right)}}{2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-48\right)}}{2}$$

Whakareatia -4 ki te -48.

$$x=\frac{-\left(-8\right)±\sqrt{64+192}}{2}$$

Tāpiri 64 ki te 192.

$$x=\frac{-\left(-8\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-8\right)±16}{2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±16}{2}$$

Nā, me whakaoti te whārite x=\frac{8±16}{2} ina he tāpiri te ±. Tāpiri 8 ki te 16.

$$x=\frac{24}{2}$$

Whakawehe 24 ki te 2.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{8±16}{2} ina he tango te ±. Tango 16 mai i 8.

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

Whakawehe -8 ki te 2.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=12 x=-4$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 8.

$$x^{2}+8=8x+56$$

Tangohia te 8x mai i ngā taha e rua.

$$x^{2}+8-8x=56$$

Tangohia te 8 mai i ngā taha e rua.

$$x^{2}-8x=56-8$$

Tangohia te 8 i te 56, ka 48.

$$x^{2}-8x=48$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=48+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=48+16$$

Tāpiri 48 ki te 16.

$$x^{2}-8x+16=64$$

Tauwehea x^{2}-8x+16. 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}.

$$\left(x-4\right)^{2}=64$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$x-4=8 x-4=-8$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=12 x=-4$$