Whakaoti i te 4x^2+3x+1=3-4x

$$4 x ^ { 2 } + 3 x + 1 = 3 - 4 x$$

Whakaoti mō x

$$x=-2$$

$$x=\frac{1}{4}=0.25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3 mai i ngā taha e rua.

$$4x^{2}+3x+1-3=-4x$$

Tangohia te 3 i te 1, ka -2.

$$4x^{2}+3x-2=-4x$$

Me tāpiri te 4x ki ngā taha e rua.

$$4x^{2}+3x-2+4x=0$$

Pahekotia te 3x me 4x, ka 7x.

$$4x^{2}+7x-2=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=4\left(-2\right)=-8$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-1 b=8$$

Tuhia anō te 4x^{2}+7x-2 hei \left(4x^{2}-x\right)+\left(8x-2\right).

$$\left(4x^{2}-x\right)+\left(8x-2\right)$$

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

$$x\left(4x-1\right)+2\left(4x-1\right)$$

Whakatauwehea atu te kīanga pātahi 4x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-1\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 4x-1=0 me te x+2=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3 mai i ngā taha e rua.

$$4x^{2}+3x+1-3=-4x$$

Tangohia te 3 i te 1, ka -2.

$$4x^{2}+3x-2=-4x$$

Me tāpiri te 4x ki ngā taha e rua.

$$4x^{2}+3x-2+4x=0$$

Pahekotia te 3x me 4x, ka 7x.

$$4x^{2}+7x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 7 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 4\left(-2\right)}}{2\times 4}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 4\left(-2\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-7±\sqrt{49-16\left(-2\right)}}{2\times 4}$$

Whakareatia -16 ki te -2.

$$x=\frac{-7±\sqrt{49+32}}{2\times 4}$$

Tāpiri 49 ki te 32.

$$x=\frac{-7±\sqrt{81}}{2\times 4}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-7±9}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-7±9}{8}$$

Nā, me whakaoti te whārite x=\frac{-7±9}{8} ina he tāpiri te ±. Tāpiri -7 ki te 9.

$$x=\frac{2}{8}$$

Whakahekea te hautanga \frac{2}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±9}{8} ina he tango te ±. Tango 9 mai i -7.

$$x=-\frac{16}{8}$$

Whakawehe -16 ki te 8.

$$x=-2$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 4x ki ngā taha e rua.

$$4x^{2}+3x+1+4x=3$$

Pahekotia te 3x me 4x, ka 7x.

$$4x^{2}+7x+1=3$$

Tangohia te 1 mai i ngā taha e rua.

$$4x^{2}+7x=3-1$$

Tangohia te 1 i te 3, ka 2.

$$4x^{2}+7x=2$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+7x}{4}=\frac{2}{4}$$

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

$$x^{2}+\frac{7}{4}x=\frac{2}{4}$$

Whakahekea te hautanga \frac{2}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{7}{4}x=\frac{1}{2}$$

Whakawehea te \frac{7}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{8}. Nā, tāpiria te pūrua o te \frac{7}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\frac{1}{2}+\left(\frac{7}{8}\right)^{2}$$

Pūruatia \frac{7}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{1}{2}+\frac{49}{64}$$

Tāpiri \frac{1}{2} ki te \frac{49}{64} 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.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{81}{64}$$

Tauwehea x^{2}+\frac{7}{4}x+\frac{49}{64}. 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}.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{81}{64}$$

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

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Whakarūnātia.

$$x+\frac{7}{8}=\frac{9}{8} x+\frac{7}{8}=-\frac{9}{8}$$

Me tango \frac{7}{8} mai i ngā taha e rua o te whārite.

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