Whakaoti i te (x-1)^2-11=25

$$( x - 1 ) ^ { 2 } - 11 = 25$$

Whakaoti mō x

$$x=7$$

$$x=-5$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-1\right)^{2}.

$$x^{2}-2x+1-11=25$$

Tangohia te 11 i te 1, ka -10.

$$x^{2}-2x-10=25$$

Tangohia te 25 mai i ngā taha e rua.

$$x^{2}-2x-10-25=0$$

Tangohia te 25 i te -10, ka -35.

$$x^{2}-2x-35=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-2x-35 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.

$$a+b=-2 ab=-35$$

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 -35.

$$1,-35 5,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-35=-34 5-7=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-7 b=5$$

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

$$\left(x-7\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+5=0.

$$x=7 x=-5$$

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-1\right)^{2}.

$$x^{2}-2x+1-11=25$$

Tangohia te 11 i te 1, ka -10.

$$x^{2}-2x-10=25$$

Tangohia te 25 mai i ngā taha e rua.

$$x^{2}-2x-10-25=0$$

Tangohia te 25 i te -10, ka -35.

$$x^{2}-2x-35=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-35\right)=-35$$

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 -35.

$$1,-35 5,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-35=-34 5-7=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-7 b=5$$

Tuhia anō te x^{2}-2x-35 hei \left(x^{2}-7x\right)+\left(5x-35\right).

$$\left(x^{2}-7x\right)+\left(5x-35\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(x-7\right)+5\left(x-7\right)$$

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

$$\left(x-7\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+5=0.

$$x=7 x=-5$$

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-1\right)^{2}.

$$x^{2}-2x+1-11=25$$

Tangohia te 11 i te 1, ka -10.

$$x^{2}-2x-10=25$$

Tangohia te 25 mai i ngā taha e rua.

$$x^{2}-2x-10-25=0$$

Tangohia te 25 i te -10, ka -35.

$$x^{2}-2x-35=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -2 mō b, me -35 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-35\right)}}{2}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-35\right)}}{2}$$

Whakareatia -4 ki te -35.

$$x=\frac{-\left(-2\right)±\sqrt{4+140}}{2}$$

Tāpiri 4 ki te 140.

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

Tuhia te pūtakerua o te 144.

$$x=\frac{-\left(-2\right)±12}{2}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±12}{2}$$

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

$$x=\frac{14}{2}$$

Whakawehe 14 ki te 2.

$$x=7$$

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

$$x=-\frac{10}{2}$$

Whakawehe -10 ki te 2.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=7 x=-5$$

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-1\right)^{2}.

$$x^{2}-2x+1-11=25$$

Tangohia te 11 i te 1, ka -10.

$$x^{2}-2x-10=25$$

Me tāpiri te 10 ki ngā taha e rua.

$$x^{2}-2x=25+10$$

Tāpirihia te 25 ki te 10, ka 35.

$$x^{2}-2x=35$$

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.

$$x^{2}-2x+1=35+1$$

Tāpiri 35 ki te 1.

$$x^{2}-2x+1=36$$

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}.

$$\left(x-1\right)^{2}=36$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{36}$$

Whakarūnātia.

$$x-1=6 x-1=-6$$

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

$$x=7 x=-5$$