Whakaoti i te 7z^2+8z+3=3z^2

$$7 z ^ { 2 } + 8 z + 3 = 3 z ^ { 2 }$$

Whakaoti mō z

$$z = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$z=-\frac{1}{2}=-0.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3z^{2} mai i ngā taha e rua.

$$7z^{2}+8z+3-3z^{2}=0$$

Pahekotia te 7z^{2} me -3z^{2}, ka 4z^{2}.

$$4z^{2}+8z+3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4z^{2}+az+bz+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=8 ab=4\times 3=12$$

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 12.

$$1,12 2,6 3,4$$

Tātaihia te tapeke mō ia takirua.

$$1+12=13 2+6=8 3+4=7$$

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

$$a=2 b=6$$

Tuhia anō te 4z^{2}+8z+3 hei \left(4z^{2}+2z\right)+\left(6z+3\right).

$$\left(4z^{2}+2z\right)+\left(6z+3\right)$$

Tauwehea te 2z i te tuatahi me te 3 i te rōpū tuarua.

$$2z\left(2z+1\right)+3\left(2z+1\right)$$

Whakatauwehea atu te kīanga pātahi 2z+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2z+1\right)\left(2z+3\right)$$

Hei kimi otinga whārite, me whakaoti te 2z+1=0 me te 2z+3=0.

$$z=-\frac{1}{2} z=-\frac{3}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3z^{2} mai i ngā taha e rua.

$$7z^{2}+8z+3-3z^{2}=0$$

Pahekotia te 7z^{2} me -3z^{2}, ka 4z^{2}.

$$4z^{2}+8z+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 8 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-8±\sqrt{8^{2}-4\times 4\times 3}}{2\times 4}$$

Pūrua 8.

$$z=\frac{-8±\sqrt{64-4\times 4\times 3}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$z=\frac{-8±\sqrt{64-16\times 3}}{2\times 4}$$

Whakareatia -16 ki te 3.

$$z=\frac{-8±\sqrt{64-48}}{2\times 4}$$

Tāpiri 64 ki te -48.

$$z=\frac{-8±\sqrt{16}}{2\times 4}$$

Tuhia te pūtakerua o te 16.

$$z=\frac{-8±4}{2\times 4}$$

Whakareatia 2 ki te 4.

$$z=\frac{-8±4}{8}$$

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

$$z=-\frac{4}{8}$$

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

$$z=-\frac{1}{2}$$

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

$$z=-\frac{12}{8}$$

Whakahekea te hautanga \frac{-12}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$z=-\frac{3}{2}$$

Kua oti te whārite te whakatau.

$$z=-\frac{1}{2} z=-\frac{3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3z^{2} mai i ngā taha e rua.

$$7z^{2}+8z+3-3z^{2}=0$$

Pahekotia te 7z^{2} me -3z^{2}, ka 4z^{2}.

$$4z^{2}+8z+3=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4z^{2}+8z=-3$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4z^{2}+8z}{4}=-\frac{3}{4}$$

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

$$z^{2}+\frac{8}{4}z=-\frac{3}{4}$$

Whakawehe 8 ki te 4.

$$z^{2}+2z=-\frac{3}{4}$$

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}+2z+1^{2}=-\frac{3}{4}+1^{2}$$

Pūrua 1.

$$z^{2}+2z+1=-\frac{3}{4}+1$$

Tāpiri -\frac{3}{4} ki te 1.

$$z^{2}+2z+1=\frac{1}{4}$$

Tauwehea z^{2}+2z+1. 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}.

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

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

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

Whakarūnātia.

$$z+1=\frac{1}{2} z+1=-\frac{1}{2}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$z=-\frac{1}{2} z=-\frac{3}{2}$$