Whakaoti i te 9a^2+24a+16=0

$$9 a ^ { 2 } + 24 a + 16 = 0$$

Whakaoti mō a

$$a = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

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 9a^{2}+aa+ba+16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=24 ab=9\times 16=144$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Tātaihia te tapeke mō ia takirua.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

Ko te otinga te takirua ka hoatu i te tapeke 24.

$$a=12 b=12$$

Tuhia anō te 9a^{2}+24a+16 hei \left(9a^{2}+12a\right)+\left(12a+16\right).

$$\left(9a^{2}+12a\right)+\left(12a+16\right)$$

Tauwehea te 3a i te tuatahi me te 4 i te rōpū tuarua.

$$3a\left(3a+4\right)+4\left(3a+4\right)$$

Whakatauwehea atu te kīanga pātahi 3a+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3a+4\right)\left(3a+4\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(3a+4\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te 3a+4=0.

$$a=-\frac{4}{3}$$

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.

$$9a^{2}+24a+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 24 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-24±\sqrt{24^{2}-4\times 9\times 16}}{2\times 9}$$

Pūrua 24.

$$a=\frac{-24±\sqrt{576-4\times 9\times 16}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$a=\frac{-24±\sqrt{576-36\times 16}}{2\times 9}$$

Whakareatia -36 ki te 16.

$$a=\frac{-24±\sqrt{576-576}}{2\times 9}$$

Tāpiri 576 ki te -576.

$$a=\frac{-24±\sqrt{0}}{2\times 9}$$

Tuhia te pūtakerua o te 0.

$$a=-\frac{24}{2\times 9}$$

Whakareatia 2 ki te 9.

$$a=-\frac{24}{18}$$

Whakahekea te hautanga \frac{-24}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$a=-\frac{4}{3}$$

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.

$$9a^{2}+24a+16=0$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$9a^{2}+24a+16-16=-16$$

Mā te tango i te 16 i a ia ake anō ka toe ko te 0.

$$9a^{2}+24a=-16$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9a^{2}+24a}{9}=-\frac{16}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$a^{2}+\frac{24}{9}a=-\frac{16}{9}$$

Whakahekea te hautanga \frac{24}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$a^{2}+\frac{8}{3}a=-\frac{16}{9}$$

Whakawehea te \frac{8}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{4}{3}. Nā, tāpiria te pūrua o te \frac{4}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+\frac{8}{3}a+\left(\frac{4}{3}\right)^{2}=-\frac{16}{9}+\left(\frac{4}{3}\right)^{2}$$

Pūruatia \frac{4}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$a^{2}+\frac{8}{3}a+\frac{16}{9}=\frac{-16+16}{9}$$

Tāpiri -\frac{16}{9} ki te \frac{16}{9} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$a^{2}+\frac{8}{3}a+\frac{16}{9}=0$$

Tauwehea a^{2}+\frac{8}{3}a+\frac{16}{9}. 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}.

$$\left(a+\frac{4}{3}\right)^{2}=0$$

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

$$\sqrt{\left(a+\frac{4}{3}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$a+\frac{4}{3}=0 a+\frac{4}{3}=0$$

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

$$a=-\frac{4}{3} a=-\frac{4}{3}$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$a=-\frac{4}{3}$$