Whakaoti i te a^2+8a-9=96

$$a ^ { 2 } + 8 a - 9 = 96$$

Whakaoti mō a

$$a=-15$$

$$a=7$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 96 mai i ngā taha e rua.

$$a^{2}+8a-9-96=0$$

Tangohia te 96 i te -9, ka -105.

$$a^{2}+8a-105=0$$

Hei whakaoti i te whārite, whakatauwehea te a^{2}+8a-105 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.

$$a+b=8 ab=-105$$

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 -105.

$$-1,105 -3,35 -5,21 -7,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

Ko te otinga te takirua ka hoatu i te tapeke 8.

$$a=-7 b=15$$

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

$$\left(a-7\right)\left(a+15\right)$$

Hei kimi otinga whārite, me whakaoti te a-7=0 me te a+15=0.

$$a=7 a=-15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 96 mai i ngā taha e rua.

$$a^{2}+8a-9-96=0$$

Tangohia te 96 i te -9, ka -105.

$$a^{2}+8a-105=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei a^{2}+aa+ba-105. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=8 ab=1\left(-105\right)=-105$$

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 -105.

$$-1,105 -3,35 -5,21 -7,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

Ko te otinga te takirua ka hoatu i te tapeke 8.

$$a=-7 b=15$$

Tuhia anō te a^{2}+8a-105 hei \left(a^{2}-7a\right)+\left(15a-105\right).

$$\left(a^{2}-7a\right)+\left(15a-105\right)$$

Tauwehea te a i te tuatahi me te 15 i te rōpū tuarua.

$$a\left(a-7\right)+15\left(a-7\right)$$

Whakatauwehea atu te kīanga pātahi a-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(a-7\right)\left(a+15\right)$$

Hei kimi otinga whārite, me whakaoti te a-7=0 me te a+15=0.

$$a=7 a=-15$$

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.

$$a^{2}+8a-9=96$$

Me tango 96 mai i ngā taha e rua o te whārite.

$$a^{2}+8a-9-96=96-96$$

Mā te tango i te 96 i a ia ake anō ka toe ko te 0.

$$a^{2}+8a-9-96=0$$

Tango 96 mai i -9.

$$a^{2}+8a-105=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 8 mō b, me -105 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-8±\sqrt{8^{2}-4\left(-105\right)}}{2}$$

Pūrua 8.

$$a=\frac{-8±\sqrt{64-4\left(-105\right)}}{2}$$

Whakareatia -4 ki te -105.

$$a=\frac{-8±\sqrt{64+420}}{2}$$

Tāpiri 64 ki te 420.

$$a=\frac{-8±\sqrt{484}}{2}$$

Tuhia te pūtakerua o te 484.

$$a=\frac{-8±22}{2}$$

Nā, me whakaoti te whārite a=\frac{-8±22}{2} ina he tāpiri te ±. Tāpiri -8 ki te 22.

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

Whakawehe 14 ki te 2.

$$a=7$$

Nā, me whakaoti te whārite a=\frac{-8±22}{2} ina he tango te ±. Tango 22 mai i -8.

$$a=-\frac{30}{2}$$

Whakawehe -30 ki te 2.

$$a=-15$$

Kua oti te whārite te whakatau.

$$a=7 a=-15$$

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.

$$a^{2}+8a-9=96$$

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

$$a^{2}+8a-9-\left(-9\right)=96-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$a^{2}+8a=96-\left(-9\right)$$

Tango -9 mai i 96.

$$a^{2}+8a=105$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+8a+4^{2}=105+4^{2}$$

Pūrua 4.

$$a^{2}+8a+16=105+16$$

Tāpiri 105 ki te 16.

$$a^{2}+8a+16=121$$

Tauwehea a^{2}+8a+16. 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}.

$$\left(a+4\right)^{2}=121$$

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

$$\sqrt{\left(a+4\right)^{2}}=\sqrt{121}$$

Whakarūnātia.

$$a+4=11 a+4=-11$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$a=7 a=-15$$