Whakaoti i te 12x^2+13x+3=0

$$12 { x }^{ 2 } +13x+3 = 0$$

Whakaoti mō x

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

$$x=-\frac{1}{3}\approx -0.333333333$$

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 12x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=12\times 3=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Tātaihia te tapeke mō ia takirua.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=4 b=9$$

Tuhia anō te 12x^{2}+13x+3 hei \left(12x^{2}+4x\right)+\left(9x+3\right).

$$\left(12x^{2}+4x\right)+\left(9x+3\right)$$

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

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

Whakatauwehea atu te kīanga pātahi 3x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

$$12x^{2}+13x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 12 mō a, 13 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 12\times 3}}{2\times 12}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\times 12\times 3}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-13±\sqrt{169-48\times 3}}{2\times 12}$$

Whakareatia -48 ki te 3.

$$x=\frac{-13±\sqrt{169-144}}{2\times 12}$$

Tāpiri 169 ki te -144.

$$x=\frac{-13±\sqrt{25}}{2\times 12}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-13±5}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{-13±5}{24}$$

Nā, me whakaoti te whārite x=\frac{-13±5}{24} ina he tāpiri te ±. Tāpiri -13 ki te 5.

$$x=-\frac{8}{24}$$

Whakahekea te hautanga \frac{-8}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Nā, me whakaoti te whārite x=\frac{-13±5}{24} ina he tango te ±. Tango 5 mai i -13.

$$x=-\frac{18}{24}$$

Whakahekea te hautanga \frac{-18}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Kua oti te whārite te whakatau.

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

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.

$$12x^{2}+13x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$12x^{2}+13x+3-3=-3$$

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

$$12x^{2}+13x=-3$$

Whakawehea ngā taha e rua ki te 12.

$$\frac{12x^{2}+13x}{12}=-\frac{3}{12}$$

Mā te whakawehe ki te 12 ka wetekia te whakareanga ki te 12.

$$x^{2}+\frac{13}{12}x=-\frac{3}{12}$$

Whakahekea te hautanga \frac{-3}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{13}{12}x=-\frac{1}{4}$$

Whakawehea te \frac{13}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{24}. Nā, tāpiria te pūrua o te \frac{13}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{13}{12}x+\left(\frac{13}{24}\right)^{2}=-\frac{1}{4}+\left(\frac{13}{24}\right)^{2}$$

Pūruatia \frac{13}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{13}{12}x+\frac{169}{576}=-\frac{1}{4}+\frac{169}{576}$$

Tāpiri -\frac{1}{4} ki te \frac{169}{576} 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.

$$x^{2}+\frac{13}{12}x+\frac{169}{576}=\frac{25}{576}$$

Tauwehea x^{2}+\frac{13}{12}x+\frac{169}{576}. 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}.

$$\left(x+\frac{13}{24}\right)^{2}=\frac{25}{576}$$

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

$$\sqrt{\left(x+\frac{13}{24}\right)^{2}}=\sqrt{\frac{25}{576}}$$

Whakarūnātia.

$$x+\frac{13}{24}=\frac{5}{24} x+\frac{13}{24}=-\frac{5}{24}$$

Me tango \frac{13}{24} mai i ngā taha e rua o te whārite.

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