Whakaoti i te 4s^2+32s+63=0

$$4 s ^ { 2 } + 32 s + 63 = 0$$

Whakaoti mō s

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

$$s = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

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 4s^{2}+as+bs+63. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=32 ab=4\times 63=252$$

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 252.

$$1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18$$

Tātaihia te tapeke mō ia takirua.

$$1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32$$

Ko te otinga te takirua ka hoatu i te tapeke 32.

$$a=14 b=18$$

Tuhia anō te 4s^{2}+32s+63 hei \left(4s^{2}+14s\right)+\left(18s+63\right).

$$\left(4s^{2}+14s\right)+\left(18s+63\right)$$

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

$$2s\left(2s+7\right)+9\left(2s+7\right)$$

Whakatauwehea atu te kīanga pātahi 2s+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2s+7\right)\left(2s+9\right)$$

Hei kimi otinga whārite, me whakaoti te 2s+7=0 me te 2s+9=0.

$$s=-\frac{7}{2} s=-\frac{9}{2}$$

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.

$$4s^{2}+32s+63=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 32 mō b, me 63 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-32±\sqrt{32^{2}-4\times 4\times 63}}{2\times 4}$$

Pūrua 32.

$$s=\frac{-32±\sqrt{1024-4\times 4\times 63}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$s=\frac{-32±\sqrt{1024-16\times 63}}{2\times 4}$$

Whakareatia -16 ki te 63.

$$s=\frac{-32±\sqrt{1024-1008}}{2\times 4}$$

Tāpiri 1024 ki te -1008.

$$s=\frac{-32±\sqrt{16}}{2\times 4}$$

Tuhia te pūtakerua o te 16.

$$s=\frac{-32±4}{2\times 4}$$

Whakareatia 2 ki te 4.

$$s=\frac{-32±4}{8}$$

Nā, me whakaoti te whārite s=\frac{-32±4}{8} ina he tāpiri te ±. Tāpiri -32 ki te 4.

$$s=-\frac{28}{8}$$

Whakahekea te hautanga \frac{-28}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$s=-\frac{7}{2}$$

Nā, me whakaoti te whārite s=\frac{-32±4}{8} ina he tango te ±. Tango 4 mai i -32.

$$s=-\frac{36}{8}$$

Whakahekea te hautanga \frac{-36}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

$$s=-\frac{7}{2} s=-\frac{9}{2}$$

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.

$$4s^{2}+32s+63=0$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$4s^{2}+32s+63-63=-63$$

Mā te tango i te 63 i a ia ake anō ka toe ko te 0.

$$4s^{2}+32s=-63$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4s^{2}+32s}{4}=-\frac{63}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$s^{2}+\frac{32}{4}s=-\frac{63}{4}$$

Whakawehe 32 ki te 4.

$$s^{2}+8s=-\frac{63}{4}$$

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.

$$s^{2}+8s+4^{2}=-\frac{63}{4}+4^{2}$$

Pūrua 4.

$$s^{2}+8s+16=-\frac{63}{4}+16$$

Tāpiri -\frac{63}{4} ki te 16.

$$s^{2}+8s+16=\frac{1}{4}$$

Tauwehea s^{2}+8s+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}.

$$\left(s+4\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(s+4\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$s+4=\frac{1}{2} s+4=-\frac{1}{2}$$

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

$$s=-\frac{7}{2} s=-\frac{9}{2}$$