Whakaoti i te 5+(c-1)^2+5+(c+4)^2=35

$$5 + ( c - 1 ) ^ { 2 } + 5 + ( c + 4 ) ^ { 2 } = 35$$

Whakaoti mō c

$$c=-4$$

$$c=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(c-1\right)^{2}.

$$5+c^{2}-2c+1+5+\left(c+4\right)^{2}=35$$

Tāpirihia te 5 ki te 1, ka 6.

$$6+c^{2}-2c+5+\left(c+4\right)^{2}=35$$

Tāpirihia te 6 ki te 5, ka 11.

$$11+c^{2}-2c+\left(c+4\right)^{2}=35$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(c+4\right)^{2}.

$$11+c^{2}-2c+c^{2}+8c+16=35$$

Pahekotia te c^{2} me c^{2}, ka 2c^{2}.

$$11+2c^{2}-2c+8c+16=35$$

Pahekotia te -2c me 8c, ka 6c.

$$11+2c^{2}+6c+16=35$$

Tāpirihia te 11 ki te 16, ka 27.

$$27+2c^{2}+6c=35$$

Tangohia te 35 mai i ngā taha e rua.

$$27+2c^{2}+6c-35=0$$

Tangohia te 35 i te 27, ka -8.

$$-8+2c^{2}+6c=0$$

Whakawehea ngā taha e rua ki te 2.

$$-4+c^{2}+3c=0$$

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

$$c^{2}+3c-4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei c^{2}+ac+bc-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=1\left(-4\right)=-4$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -4.

$$-1,4 -2,2$$

Tātaihia te tapeke mō ia takirua.

$$-1+4=3 -2+2=0$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-1 b=4$$

Tuhia anō te c^{2}+3c-4 hei \left(c^{2}-c\right)+\left(4c-4\right).

$$\left(c^{2}-c\right)+\left(4c-4\right)$$

Tauwehea te c i te tuatahi me te 4 i te rōpū tuarua.

$$c\left(c-1\right)+4\left(c-1\right)$$

Whakatauwehea atu te kīanga pātahi c-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(c-1\right)\left(c+4\right)$$

Hei kimi otinga whārite, me whakaoti te c-1=0 me te c+4=0.

$$c=1 c=-4$$

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(c-1\right)^{2}.

$$5+c^{2}-2c+1+5+\left(c+4\right)^{2}=35$$

Tāpirihia te 5 ki te 1, ka 6.

$$6+c^{2}-2c+5+\left(c+4\right)^{2}=35$$

Tāpirihia te 6 ki te 5, ka 11.

$$11+c^{2}-2c+\left(c+4\right)^{2}=35$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(c+4\right)^{2}.

$$11+c^{2}-2c+c^{2}+8c+16=35$$

Pahekotia te c^{2} me c^{2}, ka 2c^{2}.

$$11+2c^{2}-2c+8c+16=35$$

Pahekotia te -2c me 8c, ka 6c.

$$11+2c^{2}+6c+16=35$$

Tāpirihia te 11 ki te 16, ka 27.

$$27+2c^{2}+6c=35$$

Tangohia te 35 mai i ngā taha e rua.

$$27+2c^{2}+6c-35=0$$

Tangohia te 35 i te 27, ka -8.

$$-8+2c^{2}+6c=0$$

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.

$$2c^{2}+6c-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 6 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-6±\sqrt{6^{2}-4\times 2\left(-8\right)}}{2\times 2}$$

Pūrua 6.

$$c=\frac{-6±\sqrt{36-4\times 2\left(-8\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$c=\frac{-6±\sqrt{36-8\left(-8\right)}}{2\times 2}$$

Whakareatia -8 ki te -8.

$$c=\frac{-6±\sqrt{36+64}}{2\times 2}$$

Tāpiri 36 ki te 64.

$$c=\frac{-6±\sqrt{100}}{2\times 2}$$

Tuhia te pūtakerua o te 100.

$$c=\frac{-6±10}{2\times 2}$$

Whakareatia 2 ki te 2.

$$c=\frac{-6±10}{4}$$

Nā, me whakaoti te whārite c=\frac{-6±10}{4} ina he tāpiri te ±. Tāpiri -6 ki te 10.

$$c=\frac{4}{4}$$

Whakawehe 4 ki te 4.

$$c=1$$

Nā, me whakaoti te whārite c=\frac{-6±10}{4} ina he tango te ±. Tango 10 mai i -6.

$$c=-\frac{16}{4}$$

Whakawehe -16 ki te 4.

$$c=-4$$

Kua oti te whārite te whakatau.

$$c=1 c=-4$$

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(c-1\right)^{2}.

$$5+c^{2}-2c+1+5+\left(c+4\right)^{2}=35$$

Tāpirihia te 5 ki te 1, ka 6.

$$6+c^{2}-2c+5+\left(c+4\right)^{2}=35$$

Tāpirihia te 6 ki te 5, ka 11.

$$11+c^{2}-2c+\left(c+4\right)^{2}=35$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(c+4\right)^{2}.

$$11+c^{2}-2c+c^{2}+8c+16=35$$

Pahekotia te c^{2} me c^{2}, ka 2c^{2}.

$$11+2c^{2}-2c+8c+16=35$$

Pahekotia te -2c me 8c, ka 6c.

$$11+2c^{2}+6c+16=35$$

Tāpirihia te 11 ki te 16, ka 27.

$$27+2c^{2}+6c=35$$

Tangohia te 27 mai i ngā taha e rua.

$$2c^{2}+6c=35-27$$

Tangohia te 27 i te 35, ka 8.

$$2c^{2}+6c=8$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2c^{2}+6c}{2}=\frac{8}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$c^{2}+\frac{6}{2}c=\frac{8}{2}$$

Whakawehe 6 ki te 2.

$$c^{2}+3c=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$c^{2}+3c=4$$

Whakawehea te 3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{2}. Nā, tāpiria te pūrua o te \frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$c^{2}+3c+\left(\frac{3}{2}\right)^{2}=4+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$c^{2}+3c+\frac{9}{4}=4+\frac{9}{4}$$

Tāpiri 4 ki te \frac{9}{4}.

$$c^{2}+3c+\frac{9}{4}=\frac{25}{4}$$

Tauwehea c^{2}+3c+\frac{9}{4}. 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}.

$$\left(c+\frac{3}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(c+\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$c+\frac{3}{2}=\frac{5}{2} c+\frac{3}{2}=-\frac{5}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$c=1 c=-4$$