Whakaoti i te 0=4x^2-x-3

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

Whakaoti mō x

$$x=-\frac{3}{4}=-0.75$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

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

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

$$a+b=-1 ab=4\left(-3\right)=-12$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-12=-11 2-6=-4 3-4=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-4 b=3$$

Tuhia anō te 4x^{2}-x-3 hei \left(4x^{2}-4x\right)+\left(3x-3\right).

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

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

$$4x\left(x-1\right)+3\left(x-1\right)$$

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

$$\left(x-1\right)\left(4x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 4x+3=0.

$$x=1 x=-\frac{3}{4}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -1 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 4\left(-3\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-1\right)±\sqrt{1-16\left(-3\right)}}{2\times 4}$$

Whakareatia -16 ki te -3.

$$x=\frac{-\left(-1\right)±\sqrt{1+48}}{2\times 4}$$

Tāpiri 1 ki te 48.

$$x=\frac{-\left(-1\right)±\sqrt{49}}{2\times 4}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-1\right)±7}{2\times 4}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±7}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{1±7}{8}$$

Nā, me whakaoti te whārite x=\frac{1±7}{8} ina he tāpiri te ±. Tāpiri 1 ki te 7.

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

Whakawehe 8 ki te 8.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{1±7}{8} ina he tango te ±. Tango 7 mai i 1.

$$x=-\frac{6}{8}$$

Whakahekea te hautanga \frac{-6}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

$$x=1 x=-\frac{3}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

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

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-x}{4}=\frac{3}{4}$$

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

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

Whakawehea te -\frac{1}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{8}. Nā, tāpiria te pūrua o te -\frac{1}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{4}x+\left(-\frac{1}{8}\right)^{2}=\frac{3}{4}+\left(-\frac{1}{8}\right)^{2}$$

Pūruatia -\frac{1}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{3}{4}+\frac{1}{64}$$

Tāpiri \frac{3}{4} ki te \frac{1}{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.

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

Tauwehea x^{2}-\frac{1}{4}x+\frac{1}{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}.

$$\left(x-\frac{1}{8}\right)^{2}=\frac{49}{64}$$

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

$$\sqrt{\left(x-\frac{1}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Whakarūnātia.

$$x-\frac{1}{8}=\frac{7}{8} x-\frac{1}{8}=-\frac{7}{8}$$

Me tāpiri \frac{1}{8} ki ngā taha e rua o te whārite.

$$x=1 x=-\frac{3}{4}$$