Whakaoti i te (8-x)^2=25

$$( 8 - x ) ^ { 2 } = 25$$

Whakaoti mō x

$$x=3$$

$$x=13$$

Ngā Upane Whakamahi Tauwehe

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(8-x\right)^{2}.

$$64-16x+x^{2}=25$$

Tangohia te 25 mai i ngā taha e rua.

$$64-16x+x^{2}-25=0$$

Tangohia te 25 i te 64, ka 39.

$$39-16x+x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-16x+39=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-16x+39 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.

$$a+b=-16 ab=39$$

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 39.

$$-1,-39 -3,-13$$

Tātaihia te tapeke mō ia takirua.

$$-1-39=-40 -3-13=-16$$

Ko te otinga te takirua ka hoatu i te tapeke -16.

$$a=-13 b=-3$$

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

$$\left(x-13\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x-3=0.

$$x=13 x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(8-x\right)^{2}.

$$64-16x+x^{2}=25$$

Tangohia te 25 mai i ngā taha e rua.

$$64-16x+x^{2}-25=0$$

Tangohia te 25 i te 64, ka 39.

$$39-16x+x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-16x+39=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+39. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-16 ab=1\times 39=39$$

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 39.

$$-1,-39 -3,-13$$

Tātaihia te tapeke mō ia takirua.

$$-1-39=-40 -3-13=-16$$

Ko te otinga te takirua ka hoatu i te tapeke -16.

$$a=-13 b=-3$$

Tuhia anō te x^{2}-16x+39 hei \left(x^{2}-13x\right)+\left(-3x+39\right).

$$\left(x^{2}-13x\right)+\left(-3x+39\right)$$

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

$$x\left(x-13\right)-3\left(x-13\right)$$

Whakatauwehea atu te kīanga pātahi x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-13\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x-3=0.

$$x=13 x=3$$

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(8-x\right)^{2}.

$$64-16x+x^{2}=25$$

Tangohia te 25 mai i ngā taha e rua.

$$64-16x+x^{2}-25=0$$

Tangohia te 25 i te 64, ka 39.

$$39-16x+x^{2}=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$x^{2}-16x+39=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -16 mō b, me 39 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 39}}{2}$$

Pūrua -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 39}}{2}$$

Whakareatia -4 ki te 39.

$$x=\frac{-\left(-16\right)±\sqrt{256-156}}{2}$$

Tāpiri 256 ki te -156.

$$x=\frac{-\left(-16\right)±\sqrt{100}}{2}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-16\right)±10}{2}$$

Ko te tauaro o -16 ko 16.

$$x=\frac{16±10}{2}$$

Nā, me whakaoti te whārite x=\frac{16±10}{2} ina he tāpiri te ±. Tāpiri 16 ki te 10.

$$x=\frac{26}{2}$$

Whakawehe 26 ki te 2.

$$x=13$$

Nā, me whakaoti te whārite x=\frac{16±10}{2} ina he tango te ±. Tango 10 mai i 16.

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

Whakawehe 6 ki te 2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=13 x=3$$

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(8-x\right)^{2}.

$$64-16x+x^{2}=25$$

Tangohia te 64 mai i ngā taha e rua.

$$-16x+x^{2}=25-64$$

Tangohia te 64 i te 25, ka -39.

$$-16x+x^{2}=-39$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$x^{2}-16x=-39$$

Whakawehea te -16, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -8. Nā, tāpiria te pūrua o te -8 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-16x+\left(-8\right)^{2}=-39+\left(-8\right)^{2}$$

Pūrua -8.

$$x^{2}-16x+64=-39+64$$

Tāpiri -39 ki te 64.

$$x^{2}-16x+64=25$$

Tauwehea x^{2}-16x+64. 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}.

$$\left(x-8\right)^{2}=25$$

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

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-8=5 x-8=-5$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$x=13 x=3$$