Whakaoti i te 24x^2+6x-3=0

$$24 x ^ { 2 } + 6 x - 3 = 0$$

Whakaoti mō x

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

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 3.

$$8x^{2}+2x-1=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 8x^{2}+ax+bx-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=8\left(-1\right)=-8$$

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 2.

$$a=-2 b=4$$

Tuhia anō te 8x^{2}+2x-1 hei \left(8x^{2}-2x\right)+\left(4x-1\right).

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

Whakatauwehea atu 2x i te 8x^{2}-2x.

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

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

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

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

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

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.

$$24x^{2}+6x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 24 mō a, 6 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 24\left(-3\right)}}{2\times 24}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 24\left(-3\right)}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$x=\frac{-6±\sqrt{36-96\left(-3\right)}}{2\times 24}$$

Whakareatia -96 ki te -3.

$$x=\frac{-6±\sqrt{36+288}}{2\times 24}$$

Tāpiri 36 ki te 288.

$$x=\frac{-6±\sqrt{324}}{2\times 24}$$

Tuhia te pūtakerua o te 324.

$$x=\frac{-6±18}{2\times 24}$$

Whakareatia 2 ki te 24.

$$x=\frac{-6±18}{48}$$

Nā, me whakaoti te whārite x=\frac{-6±18}{48} ina he tāpiri te ±. Tāpiri -6 ki te 18.

$$x=\frac{12}{48}$$

Whakahekea te hautanga \frac{12}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

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

Nā, me whakaoti te whārite x=\frac{-6±18}{48} ina he tango te ±. Tango 18 mai i -6.

$$x=-\frac{24}{48}$$

Whakahekea te hautanga \frac{-24}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 24.

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

Kua oti te whārite te whakatau.

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

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.

$$24x^{2}+6x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$24x^{2}+6x-3-\left(-3\right)=-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$24x^{2}+6x=-\left(-3\right)$$

Tango -3 mai i 0.

$$24x^{2}+6x=3$$

Whakawehea ngā taha e rua ki te 24.

$$\frac{24x^{2}+6x}{24}=\frac{3}{24}$$

Mā te whakawehe ki te 24 ka wetekia te whakareanga ki te 24.

$$x^{2}+\frac{6}{24}x=\frac{3}{24}$$

Whakahekea te hautanga \frac{6}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x^{2}+\frac{1}{4}x=\frac{3}{24}$$

Whakahekea te hautanga \frac{3}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

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

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.

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

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

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{1}{8}+\frac{1}{64}$$

Tāpiri \frac{1}{8} 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.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{9}{64}$$

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}.

$$\left(x+\frac{1}{8}\right)^{2}=\frac{9}{64}$$

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

$$\sqrt{\left(x+\frac{1}{8}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Whakarūnātia.

$$x+\frac{1}{8}=\frac{3}{8} x+\frac{1}{8}=-\frac{3}{8}$$

Me tango \frac{1}{8} mai i ngā taha e rua o te whārite.

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