Whakaoti i te x^2+(x-2)^2=100

$$x ^ { 2 } + ( x - 2 ) ^ { 2 } = 100$$

Whakaoti mō x

$$x=-6$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$x^{2}+x^{2}-4x+4=100$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-4x+4=100$$

Tangohia te 100 mai i ngā taha e rua.

$$2x^{2}-4x+4-100=0$$

Tangohia te 100 i te 4, ka -96.

$$2x^{2}-4x-96=0$$

Whakawehea ngā taha e rua ki te 2.

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

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.

$$a+b=-2 ab=1\left(-48\right)=-48$$

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 -2.

$$a=-8 b=6$$

Tuhia anō te x^{2}-2x-48 hei \left(x^{2}-8x\right)+\left(6x-48\right).

$$\left(x^{2}-8x\right)+\left(6x-48\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-8\right)+6\left(x-8\right)$$

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

$$\left(x-8\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+6=0.

$$x=8 x=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$x^{2}+x^{2}-4x+4=100$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-4x+4=100$$

Tangohia te 100 mai i ngā taha e rua.

$$2x^{2}-4x+4-100=0$$

Tangohia te 100 i te 4, ka -96.

$$2x^{2}-4x-96=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -4 mō b, me -96 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-96\right)}}{2\times 2}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-96\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-96\right)}}{2\times 2}$$

Whakareatia -8 ki te -96.

$$x=\frac{-\left(-4\right)±\sqrt{16+768}}{2\times 2}$$

Tāpiri 16 ki te 768.

$$x=\frac{-\left(-4\right)±\sqrt{784}}{2\times 2}$$

Tuhia te pūtakerua o te 784.

$$x=\frac{-\left(-4\right)±28}{2\times 2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±28}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{4±28}{4}$$

Nā, me whakaoti te whārite x=\frac{4±28}{4} ina he tāpiri te ±. Tāpiri 4 ki te 28.

$$x=\frac{32}{4}$$

Whakawehe 32 ki te 4.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{4±28}{4} ina he tango te ±. Tango 28 mai i 4.

$$x=-\frac{24}{4}$$

Whakawehe -24 ki te 4.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=8 x=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$x^{2}+x^{2}-4x+4=100$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-4x+4=100$$

Tangohia te 4 mai i ngā taha e rua.

$$2x^{2}-4x=100-4$$

Tangohia te 4 i te 100, ka 96.

$$2x^{2}-4x=96$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-4x}{2}=\frac{96}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{96}{2}$$

Whakawehe -4 ki te 2.

$$x^{2}-2x=\frac{96}{2}$$

Whakawehe 96 ki te 2.

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

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=48+1$$

Tāpiri 48 ki te 1.

$$x^{2}-2x+1=49$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=49$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{49}$$

Whakarūnātia.

$$x-1=7 x-1=-7$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=8 x=-6$$