Whakaoti i te 16x^2+74x+9=0

$$16 x ^ { 2 } + 74 x + 9 = 0$$

Whakaoti mō x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=-\frac{1}{8}=-0.125$$

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 16x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=74 ab=16\times 9=144$$

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 74.

$$a=2 b=72$$

Tuhia anō te 16x^{2}+74x+9 hei \left(16x^{2}+2x\right)+\left(72x+9\right).

$$\left(16x^{2}+2x\right)+\left(72x+9\right)$$

Tauwehea te 2x i te tuatahi me te 9 i te rōpū tuarua.

$$2x\left(8x+1\right)+9\left(8x+1\right)$$

Whakatauwehea atu te kīanga pātahi 8x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(8x+1\right)\left(2x+9\right)$$

Hei kimi otinga whārite, me whakaoti te 8x+1=0 me te 2x+9=0.

$$x=-\frac{1}{8} x=-\frac{9}{2}$$

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.

$$16x^{2}+74x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, 74 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-74±\sqrt{74^{2}-4\times 16\times 9}}{2\times 16}$$

Pūrua 74.

$$x=\frac{-74±\sqrt{5476-4\times 16\times 9}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-74±\sqrt{5476-64\times 9}}{2\times 16}$$

Whakareatia -64 ki te 9.

$$x=\frac{-74±\sqrt{5476-576}}{2\times 16}$$

Tāpiri 5476 ki te -576.

$$x=\frac{-74±\sqrt{4900}}{2\times 16}$$

Tuhia te pūtakerua o te 4900.

$$x=\frac{-74±70}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{-74±70}{32}$$

Nā, me whakaoti te whārite x=\frac{-74±70}{32} ina he tāpiri te ±. Tāpiri -74 ki te 70.

$$x=-\frac{4}{32}$$

Whakahekea te hautanga \frac{-4}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{1}{8}$$

Nā, me whakaoti te whārite x=\frac{-74±70}{32} ina he tango te ±. Tango 70 mai i -74.

$$x=-\frac{144}{32}$$

Whakahekea te hautanga \frac{-144}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$x=-\frac{9}{2}$$

Kua oti te whārite te whakatau.

$$x=-\frac{1}{8} x=-\frac{9}{2}$$

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.

$$16x^{2}+74x+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$16x^{2}+74x+9-9=-9$$

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

$$16x^{2}+74x=-9$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}+74x}{16}=-\frac{9}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\frac{74}{16}x=-\frac{9}{16}$$

Whakahekea te hautanga \frac{74}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{37}{8}x=-\frac{9}{16}$$

Whakawehea te \frac{37}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{37}{16}. Nā, tāpiria te pūrua o te \frac{37}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{37}{8}x+\left(\frac{37}{16}\right)^{2}=-\frac{9}{16}+\left(\frac{37}{16}\right)^{2}$$

Pūruatia \frac{37}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{37}{8}x+\frac{1369}{256}=-\frac{9}{16}+\frac{1369}{256}$$

Tāpiri -\frac{9}{16} ki te \frac{1369}{256} 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.

$$x^{2}+\frac{37}{8}x+\frac{1369}{256}=\frac{1225}{256}$$

Tauwehea x^{2}+\frac{37}{8}x+\frac{1369}{256}. 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}.

$$\left(x+\frac{37}{16}\right)^{2}=\frac{1225}{256}$$

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

$$\sqrt{\left(x+\frac{37}{16}\right)^{2}}=\sqrt{\frac{1225}{256}}$$

Whakarūnātia.

$$x+\frac{37}{16}=\frac{35}{16} x+\frac{37}{16}=-\frac{35}{16}$$

Me tango \frac{37}{16} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{8} x=-\frac{9}{2}$$