Whakaoti i te a^2-8a-48=0

$$a ^ { 2 } - 8 a - 48 = 0$$

Whakaoti mō a

$$a=-4$$

$$a=12$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te a^{2}-8a-48 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.

$$a+b=-8 ab=-48$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

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

$$a=-12 b=4$$

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

$$\left(a-12\right)\left(a+4\right)$$

Hei kimi otinga whārite, me whakaoti te a-12=0 me te a+4=0.

$$a=12 a=-4$$

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 a^{2}+aa+ba-48. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\left(-48\right)=-48$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

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

$$a=-12 b=4$$

Tuhia anō te a^{2}-8a-48 hei \left(a^{2}-12a\right)+\left(4a-48\right).

$$\left(a^{2}-12a\right)+\left(4a-48\right)$$

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

$$a\left(a-12\right)+4\left(a-12\right)$$

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

$$\left(a-12\right)\left(a+4\right)$$

Hei kimi otinga whārite, me whakaoti te a-12=0 me te a+4=0.

$$a=12 a=-4$$

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.

$$a^{2}-8a-48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me -48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-48\right)}}{2}$$

Pūrua -8.

$$a=\frac{-\left(-8\right)±\sqrt{64-4\left(-48\right)}}{2}$$

Whakareatia -4 ki te -48.

$$a=\frac{-\left(-8\right)±\sqrt{64+192}}{2}$$

Tāpiri 64 ki te 192.

$$a=\frac{-\left(-8\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$a=\frac{-\left(-8\right)±16}{2}$$

Ko te tauaro o -8 ko 8.

$$a=\frac{8±16}{2}$$

Nā, me whakaoti te whārite a=\frac{8±16}{2} ina he tāpiri te ±. Tāpiri 8 ki te 16.

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

Whakawehe 24 ki te 2.

$$a=12$$

Nā, me whakaoti te whārite a=\frac{8±16}{2} ina he tango te ±. Tango 16 mai i 8.

$$a=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$a=-4$$

Kua oti te whārite te whakatau.

$$a=12 a=-4$$

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.

$$a^{2}-8a-48=0$$

Me tāpiri 48 ki ngā taha e rua o te whārite.

$$a^{2}-8a-48-\left(-48\right)=-\left(-48\right)$$

Mā te tango i te -48 i a ia ake anō ka toe ko te 0.

$$a^{2}-8a=-\left(-48\right)$$

Tango -48 mai i 0.

$$a^{2}-8a=48$$

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.

$$a^{2}-8a+\left(-4\right)^{2}=48+\left(-4\right)^{2}$$

Pūrua -4.

$$a^{2}-8a+16=48+16$$

Tāpiri 48 ki te 16.

$$a^{2}-8a+16=64$$

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}.

$$\left(a-4\right)^{2}=64$$

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

$$\sqrt{\left(a-4\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$a-4=8 a-4=-8$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$a=12 a=-4$$