Whakaoti i te (x-7)^2-8=17

$$( x - 7 ) ^ { 2 } - 8 = 17$$

Whakaoti mō x

$$x=12$$

$$x=2$$

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}.

$$x^{2}-14x+49-8=17$$

Tangohia te 8 i te 49, ka 41.

$$x^{2}-14x+41=17$$

Tangohia te 17 mai i ngā taha e rua.

$$x^{2}-14x+41-17=0$$

Tangohia te 17 i te 41, ka 24.

$$x^{2}-14x+24=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-14x+24 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.

$$a+b=-14 ab=24$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 24.

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

Tātaihia te tapeke mō ia takirua.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

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

$$a=-12 b=-2$$

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

$$\left(x-12\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x-2=0.

$$x=12 x=2$$

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}.

$$x^{2}-14x+49-8=17$$

Tangohia te 8 i te 49, ka 41.

$$x^{2}-14x+41=17$$

Tangohia te 17 mai i ngā taha e rua.

$$x^{2}-14x+41-17=0$$

Tangohia te 17 i te 41, ka 24.

$$x^{2}-14x+24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=1\times 24=24$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 24.

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

Tātaihia te tapeke mō ia takirua.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

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

$$a=-12 b=-2$$

Tuhia anō te x^{2}-14x+24 hei \left(x^{2}-12x\right)+\left(-2x+24\right).

$$\left(x^{2}-12x\right)+\left(-2x+24\right)$$

Tauwehea te x i te tuatahi me te -2 i te rōpū tuarua.

$$x\left(x-12\right)-2\left(x-12\right)$$

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

$$\left(x-12\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x-2=0.

$$x=12 x=2$$

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}.

$$x^{2}-14x+49-8=17$$

Tangohia te 8 i te 49, ka 41.

$$x^{2}-14x+41=17$$

Tangohia te 17 mai i ngā taha e rua.

$$x^{2}-14x+41-17=0$$

Tangohia te 17 i te 41, ka 24.

$$x^{2}-14x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -14 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 24}}{2}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 24}}{2}$$

Whakareatia -4 ki te 24.

$$x=\frac{-\left(-14\right)±\sqrt{196-96}}{2}$$

Tāpiri 196 ki te -96.

$$x=\frac{-\left(-14\right)±\sqrt{100}}{2}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-14\right)±10}{2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±10}{2}$$

Nā, me whakaoti te whārite x=\frac{14±10}{2} ina he tāpiri te ±. Tāpiri 14 ki te 10.

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

Whakawehe 24 ki te 2.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{14±10}{2} ina he tango te ±. Tango 10 mai i 14.

$$x=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=12 x=2$$

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-7\right)^{2}.

$$x^{2}-14x+49-8=17$$

Tangohia te 8 i te 49, ka 41.

$$x^{2}-14x+41=17$$

Tangohia te 41 mai i ngā taha e rua.

$$x^{2}-14x=17-41$$

Tangohia te 41 i te 17, ka -24.

$$x^{2}-14x=-24$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-24+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-24+49$$

Tāpiri -24 ki te 49.

$$x^{2}-14x+49=25$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=25$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-7=5 x-7=-5$$

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

$$x=12 x=2$$