Whakaoti i te (x-4)^2-9=0

$$( x - 4 ) ^ { 2 } - 9 = 0$$

Whakaoti mō x

$$x=7$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(x-4\right)^{2}.

$$x^{2}-8x+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$x^{2}-8x+7=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-8x+7 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.

$$a+b=-8 ab=7$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-7 b=-1$$

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

$$\left(x-7\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x-1=0.

$$x=7 x=1$$

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-4\right)^{2}.

$$x^{2}-8x+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$x^{2}-8x+7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\times 7=7$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=-7 b=-1$$

Tuhia anō te x^{2}-8x+7 hei \left(x^{2}-7x\right)+\left(-x+7\right).

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

Tauwehea te x i te tuatahi me te -1 i te rōpū tuarua.

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

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

$$\left(x-7\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x-1=0.

$$x=7 x=1$$

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-4\right)^{2}.

$$x^{2}-8x+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$x^{2}-8x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 7}}{2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 7}}{2}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-8\right)±\sqrt{64-28}}{2}$$

Tāpiri 64 ki te -28.

$$x=\frac{-\left(-8\right)±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-8\right)±6}{2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±6}{2}$$

Nā, me whakaoti te whārite x=\frac{8±6}{2} ina he tāpiri te ±. Tāpiri 8 ki te 6.

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

Whakawehe 14 ki te 2.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{8±6}{2} ina he tango te ±. Tango 6 mai i 8.

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

Whakawehe 2 ki te 2.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=7 x=1$$

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-4\right)^{2}.

$$x^{2}-8x+16-9=0$$

Tangohia te 9 i te 16, ka 7.

$$x^{2}-8x+7=0$$

Tangohia te 7 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{2}-8x=-7$$

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.

$$x^{2}-8x+\left(-4\right)^{2}=-7+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-7+16$$

Tāpiri -7 ki te 16.

$$x^{2}-8x+16=9$$

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}.

$$\left(x-4\right)^{2}=9$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x-4=3 x-4=-3$$

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

$$x=7 x=1$$