Whakaoti i te (x-7)^2=100

$$( x - 7 ) ^ { 2 } = 100$$

Whakaoti mō x

$$x=17$$

$$x=-3$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-7\right)^{2}.

$$x^{2}-14x+49=100$$

Tangohia te 100 mai i ngā taha e rua.

$$x^{2}-14x+49-100=0$$

Tangohia te 100 i te 49, ka -51.

$$x^{2}-14x-51=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-14x-51 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.

$$a+b=-14 ab=-51$$

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 -51.

$$1,-51 3,-17$$

Tātaihia te tapeke mō ia takirua.

$$1-51=-50 3-17=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-17 b=3$$

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

$$\left(x-17\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-17=0 me te x+3=0.

$$x=17 x=-3$$

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-7\right)^{2}.

$$x^{2}-14x+49=100$$

Tangohia te 100 mai i ngā taha e rua.

$$x^{2}-14x+49-100=0$$

Tangohia te 100 i te 49, ka -51.

$$x^{2}-14x-51=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-51. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=1\left(-51\right)=-51$$

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 -51.

$$1,-51 3,-17$$

Tātaihia te tapeke mō ia takirua.

$$1-51=-50 3-17=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-17 b=3$$

Tuhia anō te x^{2}-14x-51 hei \left(x^{2}-17x\right)+\left(3x-51\right).

$$\left(x^{2}-17x\right)+\left(3x-51\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(x-17\right)+3\left(x-17\right)$$

Whakatauwehea atu te kīanga pātahi x-17 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-17\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-17=0 me te x+3=0.

$$x=17 x=-3$$

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-7\right)^{2}.

$$x^{2}-14x+49=100$$

Tangohia te 100 mai i ngā taha e rua.

$$x^{2}-14x+49-100=0$$

Tangohia te 100 i te 49, ka -51.

$$x^{2}-14x-51=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -14 mō b, me -51 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-51\right)}}{2}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-51\right)}}{2}$$

Whakareatia -4 ki te -51.

$$x=\frac{-\left(-14\right)±\sqrt{196+204}}{2}$$

Tāpiri 196 ki te 204.

$$x=\frac{-\left(-14\right)±\sqrt{400}}{2}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-14\right)±20}{2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±20}{2}$$

Nā, me whakaoti te whārite x=\frac{14±20}{2} ina he tāpiri te ±. Tāpiri 14 ki te 20.

$$x=\frac{34}{2}$$

Whakawehe 34 ki te 2.

$$x=17$$

Nā, me whakaoti te whārite x=\frac{14±20}{2} ina he tango te ±. Tango 20 mai i 14.

$$x=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=17 x=-3$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{100}$$

Whakarūnātia.

$$x-7=10 x-7=-10$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=17 x=-3$$