Whakaoti i te 36=108+c^2-18c

$$36 = 108+ { c }^{ 2 } -18c$$

Whakaoti mō c

$$c=6$$

$$c=12$$

Ngā Upane Whakamahi Tauwehe

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$108+c^{2}-18c=36$$

Tangohia te 36 mai i ngā taha e rua.

$$108+c^{2}-18c-36=0$$

Tangohia te 36 i te 108, ka 72.

$$72+c^{2}-18c=0$$

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

$$c^{2}-18c+72=0$$

Hei whakaoti i te whārite, whakatauwehea te c^{2}-18c+72 mā te whakamahi i te tātai c^{2}+\left(a+b\right)c+ab=\left(c+a\right)\left(c+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-18 ab=72$$

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 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

Ko te otinga te takirua ka hoatu i te tapeke -18.

$$a=-12 b=-6$$

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

$$\left(c-12\right)\left(c-6\right)$$

Hei kimi otinga whārite, me whakaoti te c-12=0 me te c-6=0.

$$c=12 c=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$108+c^{2}-18c=36$$

Tangohia te 36 mai i ngā taha e rua.

$$108+c^{2}-18c-36=0$$

Tangohia te 36 i te 108, ka 72.

$$72+c^{2}-18c=0$$

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

$$c^{2}-18c+72=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei c^{2}+ac+bc+72. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-18 ab=1\times 72=72$$

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 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

Ko te otinga te takirua ka hoatu i te tapeke -18.

$$a=-12 b=-6$$

Tuhia anō te c^{2}-18c+72 hei \left(c^{2}-12c\right)+\left(-6c+72\right).

$$\left(c^{2}-12c\right)+\left(-6c+72\right)$$

Tauwehea te c i te tuatahi me te -6 i te rōpū tuarua.

$$c\left(c-12\right)-6\left(c-12\right)$$

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

$$\left(c-12\right)\left(c-6\right)$$

Hei kimi otinga whārite, me whakaoti te c-12=0 me te c-6=0.

$$c=12 c=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$108+c^{2}-18c=36$$

Tangohia te 36 mai i ngā taha e rua.

$$108+c^{2}-18c-36=0$$

Tangohia te 36 i te 108, ka 72.

$$72+c^{2}-18c=0$$

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.

$$c^{2}-18c+72=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -18 mō b, me 72 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 72}}{2}$$

Pūrua -18.

$$c=\frac{-\left(-18\right)±\sqrt{324-4\times 72}}{2}$$

Whakareatia -4 ki te 72.

$$c=\frac{-\left(-18\right)±\sqrt{324-288}}{2}$$

Tāpiri 324 ki te -288.

$$c=\frac{-\left(-18\right)±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$c=\frac{-\left(-18\right)±6}{2}$$

Ko te tauaro o -18 ko 18.

$$c=\frac{18±6}{2}$$

Nā, me whakaoti te whārite c=\frac{18±6}{2} ina he tāpiri te ±. Tāpiri 18 ki te 6.

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

Whakawehe 24 ki te 2.

$$c=12$$

Nā, me whakaoti te whārite c=\frac{18±6}{2} ina he tango te ±. Tango 6 mai i 18.

$$c=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$c=6$$

Kua oti te whārite te whakatau.

$$c=12 c=6$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$108+c^{2}-18c=36$$

Tangohia te 108 mai i ngā taha e rua.

$$c^{2}-18c=36-108$$

Tangohia te 108 i te 36, ka -72.

$$c^{2}-18c=-72$$

Whakawehea te -18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -9. Nā, tāpiria te pūrua o te -9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$c^{2}-18c+\left(-9\right)^{2}=-72+\left(-9\right)^{2}$$

Pūrua -9.

$$c^{2}-18c+81=-72+81$$

Tāpiri -72 ki te 81.

$$c^{2}-18c+81=9$$

Tauwehea c^{2}-18c+81. 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}.

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

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

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

Whakarūnātia.

$$c-9=3 c-9=-3$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$c=12 c=6$$