Whakaoti i te [x^2-16x+63=0]

$$[ x ^ { 2 } - 16 x + 63 = 0 ]$$

Whakaoti mō x

$$x=7$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-16x+63 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.

$$a+b=-16 ab=63$$

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 63.

$$-1,-63 -3,-21 -7,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-63=-64 -3-21=-24 -7-9=-16$$

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

$$a=-9 b=-7$$

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

$$\left(x-9\right)\left(x-7\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x-7=0.

$$x=9 x=7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+63. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-16 ab=1\times 63=63$$

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 63.

$$-1,-63 -3,-21 -7,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-63=-64 -3-21=-24 -7-9=-16$$

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

$$a=-9 b=-7$$

Tuhia anō te x^{2}-16x+63 hei \left(x^{2}-9x\right)+\left(-7x+63\right).

$$\left(x^{2}-9x\right)+\left(-7x+63\right)$$

Tauwehea te x i te tuatahi me te -7 i te rōpū tuarua.

$$x\left(x-9\right)-7\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(x-7\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x-7=0.

$$x=9 x=7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}-16x+63=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -16 mō b, me 63 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 63}}{2}$$

Pūrua -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 63}}{2}$$

Whakareatia -4 ki te 63.

$$x=\frac{-\left(-16\right)±\sqrt{256-252}}{2}$$

Tāpiri 256 ki te -252.

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

Tuhia te pūtakerua o te 4.

$$x=\frac{-\left(-16\right)±2}{2}$$

Ko te tauaro o -16 ko 16.

$$x=\frac{16±2}{2}$$

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

$$x=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$x=9$$

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

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

Whakawehe 14 ki te 2.

$$x=7$$

Kua oti te whārite te whakatau.

$$x=9 x=7$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}-16x+63=0$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$x^{2}-16x+63-63=-63$$

Mā te tango i te 63 i a ia ake anō ka toe ko te 0.

$$x^{2}-16x=-63$$

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.

$$x^{2}-16x+\left(-8\right)^{2}=-63+\left(-8\right)^{2}$$

Pūrua -8.

$$x^{2}-16x+64=-63+64$$

Tāpiri -63 ki te 64.

$$x^{2}-16x+64=1$$

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}.

$$\left(x-8\right)^{2}=1$$

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

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-8=1 x-8=-1$$

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

$$x=9 x=7$$