Whakaoti i te 2943=a^2+1/2a*41*4

$$2943 = a ^ { 2 } + \frac { 1 } { 2 } a \cdot 41 \cdot 4$$

Whakaoti mō a

$$a=-109$$

$$a=27$$

Ngā Upane Whakamahi Tauwehe

Whakareatia te \frac{1}{2} ki te 41, ka \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Whakareatia te \frac{41}{2} ki te 4, ka 82.

$$2943=a^{2}+82a$$

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

$$a^{2}+82a=2943$$

Tangohia te 2943 mai i ngā taha e rua.

$$a^{2}+82a-2943=0$$

Hei whakaoti i te whārite, whakatauwehea te a^{2}+82a-2943 mā te whakamahi i te tātai a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=82 ab=-2943$$

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 -2943.

$$-1,2943 -3,981 -9,327 -27,109$$

Tātaihia te tapeke mō ia takirua.

$$-1+2943=2942 -3+981=978 -9+327=318 -27+109=82$$

Ko te otinga te takirua ka hoatu i te tapeke 82.

$$a=-27 b=109$$

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

$$\left(a-27\right)\left(a+109\right)$$

Hei kimi otinga whārite, me whakaoti te a-27=0 me te a+109=0.

$$a=27 a=-109$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia te \frac{1}{2} ki te 41, ka \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Whakareatia te \frac{41}{2} ki te 4, ka 82.

$$2943=a^{2}+82a$$

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

$$a^{2}+82a=2943$$

Tangohia te 2943 mai i ngā taha e rua.

$$a^{2}+82a-2943=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei a^{2}+aa+ba-2943. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=82 ab=1\left(-2943\right)=-2943$$

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 -2943.

$$-1,2943 -3,981 -9,327 -27,109$$

Tātaihia te tapeke mō ia takirua.

$$-1+2943=2942 -3+981=978 -9+327=318 -27+109=82$$

Ko te otinga te takirua ka hoatu i te tapeke 82.

$$a=-27 b=109$$

Tuhia anō te a^{2}+82a-2943 hei \left(a^{2}-27a\right)+\left(109a-2943\right).

$$\left(a^{2}-27a\right)+\left(109a-2943\right)$$

Tauwehea te a i te tuatahi me te 109 i te rōpū tuarua.

$$a\left(a-27\right)+109\left(a-27\right)$$

Whakatauwehea atu te kīanga pātahi a-27 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(a-27\right)\left(a+109\right)$$

Hei kimi otinga whārite, me whakaoti te a-27=0 me te a+109=0.

$$a=27 a=-109$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te \frac{1}{2} ki te 41, ka \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Whakareatia te \frac{41}{2} ki te 4, ka 82.

$$2943=a^{2}+82a$$

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

$$a^{2}+82a=2943$$

Tangohia te 2943 mai i ngā taha e rua.

$$a^{2}+82a-2943=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 82 mō b, me -2943 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-82±\sqrt{82^{2}-4\left(-2943\right)}}{2}$$

Pūrua 82.

$$a=\frac{-82±\sqrt{6724-4\left(-2943\right)}}{2}$$

Whakareatia -4 ki te -2943.

$$a=\frac{-82±\sqrt{6724+11772}}{2}$$

Tāpiri 6724 ki te 11772.

$$a=\frac{-82±\sqrt{18496}}{2}$$

Tuhia te pūtakerua o te 18496.

$$a=\frac{-82±136}{2}$$

Nā, me whakaoti te whārite a=\frac{-82±136}{2} ina he tāpiri te ±. Tāpiri -82 ki te 136.

$$a=\frac{54}{2}$$

Whakawehe 54 ki te 2.

$$a=27$$

Nā, me whakaoti te whārite a=\frac{-82±136}{2} ina he tango te ±. Tango 136 mai i -82.

$$a=-\frac{218}{2}$$

Whakawehe -218 ki te 2.

$$a=-109$$

Kua oti te whārite te whakatau.

$$a=27 a=-109$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te \frac{1}{2} ki te 41, ka \frac{41}{2}.

$$2943=a^{2}+\frac{41}{2}a\times 4$$

Whakareatia te \frac{41}{2} ki te 4, ka 82.

$$2943=a^{2}+82a$$

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

$$a^{2}+82a=2943$$

Whakawehea te 82, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 41. Nā, tāpiria te pūrua o te 41 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+82a+41^{2}=2943+41^{2}$$

Pūrua 41.

$$a^{2}+82a+1681=2943+1681$$

Tāpiri 2943 ki te 1681.

$$a^{2}+82a+1681=4624$$

Tauwehea a^{2}+82a+1681. 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}.

$$\left(a+41\right)^{2}=4624$$

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

$$\sqrt{\left(a+41\right)^{2}}=\sqrt{4624}$$

Whakarūnātia.

$$a+41=68 a+41=-68$$

Me tango 41 mai i ngā taha e rua o te whārite.

$$a=27 a=-109$$