Whakaoti i te x^2-60x+864=0

$${ x }^{ 2 } -60x+864 = 0$$

Whakaoti mō x

$$x=24$$

$$x=36$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-60x+864 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.

$$a+b=-60 ab=864$$

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 864.

$$-1,-864 -2,-432 -3,-288 -4,-216 -6,-144 -8,-108 -9,-96 -12,-72 -16,-54 -18,-48 -24,-36 -27,-32$$

Tātaihia te tapeke mō ia takirua.

$$-1-864=-865 -2-432=-434 -3-288=-291 -4-216=-220 -6-144=-150 -8-108=-116 -9-96=-105 -12-72=-84 -16-54=-70 -18-48=-66 -24-36=-60 -27-32=-59$$

Ko te otinga te takirua ka hoatu i te tapeke -60.

$$a=-36 b=-24$$

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

$$\left(x-36\right)\left(x-24\right)$$

Hei kimi otinga whārite, me whakaoti te x-36=0 me te x-24=0.

$$x=36 x=24$$

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+864. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-60 ab=1\times 864=864$$

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 864.

$$-1,-864 -2,-432 -3,-288 -4,-216 -6,-144 -8,-108 -9,-96 -12,-72 -16,-54 -18,-48 -24,-36 -27,-32$$

Tātaihia te tapeke mō ia takirua.

$$-1-864=-865 -2-432=-434 -3-288=-291 -4-216=-220 -6-144=-150 -8-108=-116 -9-96=-105 -12-72=-84 -16-54=-70 -18-48=-66 -24-36=-60 -27-32=-59$$

Ko te otinga te takirua ka hoatu i te tapeke -60.

$$a=-36 b=-24$$

Tuhia anō te x^{2}-60x+864 hei \left(x^{2}-36x\right)+\left(-24x+864\right).

$$\left(x^{2}-36x\right)+\left(-24x+864\right)$$

Tauwehea te x i te tuatahi me te -24 i te rōpū tuarua.

$$x\left(x-36\right)-24\left(x-36\right)$$

Whakatauwehea atu te kīanga pātahi x-36 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-36\right)\left(x-24\right)$$

Hei kimi otinga whārite, me whakaoti te x-36=0 me te x-24=0.

$$x=36 x=24$$

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.

$$x^{2}-60x+864=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -60 mō b, me 864 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 864}}{2}$$

Pūrua -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 864}}{2}$$

Whakareatia -4 ki te 864.

$$x=\frac{-\left(-60\right)±\sqrt{3600-3456}}{2}$$

Tāpiri 3600 ki te -3456.

$$x=\frac{-\left(-60\right)±\sqrt{144}}{2}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-\left(-60\right)±12}{2}$$

Ko te tauaro o -60 ko 60.

$$x=\frac{60±12}{2}$$

Nā, me whakaoti te whārite x=\frac{60±12}{2} ina he tāpiri te ±. Tāpiri 60 ki te 12.

$$x=\frac{72}{2}$$

Whakawehe 72 ki te 2.

$$x=36$$

Nā, me whakaoti te whārite x=\frac{60±12}{2} ina he tango te ±. Tango 12 mai i 60.

$$x=\frac{48}{2}$$

Whakawehe 48 ki te 2.

$$x=24$$

Kua oti te whārite te whakatau.

$$x=36 x=24$$

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.

$$x^{2}-60x+864=0$$

Me tango 864 mai i ngā taha e rua o te whārite.

$$x^{2}-60x+864-864=-864$$

Mā te tango i te 864 i a ia ake anō ka toe ko te 0.

$$x^{2}-60x=-864$$

Whakawehea te -60, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -30. Nā, tāpiria te pūrua o te -30 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-60x+\left(-30\right)^{2}=-864+\left(-30\right)^{2}$$

Pūrua -30.

$$x^{2}-60x+900=-864+900$$

Tāpiri -864 ki te 900.

$$x^{2}-60x+900=36$$

Tauwehea x^{2}-60x+900. 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}.

$$\left(x-30\right)^{2}=36$$

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

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{36}$$

Whakarūnātia.

$$x-30=6 x-30=-6$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$x=36 x=24$$