Whakaoti i te 3x^2-5x-372=0

$$3 { x }^{ 2 } -5x-372=0$$

Whakaoti mō x

$$x = -\frac{31}{3} = -10\frac{1}{3} \approx -10.333333333$$

$$x=12$$

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 3x^{2}+ax+bx-372. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=3\left(-372\right)=-1116$$

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 -1116.

$$1,-1116 2,-558 3,-372 4,-279 6,-186 9,-124 12,-93 18,-62 31,-36$$

Tātaihia te tapeke mō ia takirua.

$$1-1116=-1115 2-558=-556 3-372=-369 4-279=-275 6-186=-180 9-124=-115 12-93=-81 18-62=-44 31-36=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-36 b=31$$

Tuhia anō te 3x^{2}-5x-372 hei \left(3x^{2}-36x\right)+\left(31x-372\right).

$$\left(3x^{2}-36x\right)+\left(31x-372\right)$$

Tauwehea te 3x i te tuatahi me te 31 i te rōpū tuarua.

$$3x\left(x-12\right)+31\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(3x+31\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te 3x+31=0.

$$x=12 x=-\frac{31}{3}$$

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.

$$3x^{2}-5x-372=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -5 mō b, me -372 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-372\right)}}{2\times 3}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 3\left(-372\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-372\right)}}{2\times 3}$$

Whakareatia -12 ki te -372.

$$x=\frac{-\left(-5\right)±\sqrt{25+4464}}{2\times 3}$$

Tāpiri 25 ki te 4464.

$$x=\frac{-\left(-5\right)±\sqrt{4489}}{2\times 3}$$

Tuhia te pūtakerua o te 4489.

$$x=\frac{-\left(-5\right)±67}{2\times 3}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±67}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{5±67}{6}$$

Nā, me whakaoti te whārite x=\frac{5±67}{6} ina he tāpiri te ±. Tāpiri 5 ki te 67.

$$x=\frac{72}{6}$$

Whakawehe 72 ki te 6.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{5±67}{6} ina he tango te ±. Tango 67 mai i 5.

$$x=-\frac{62}{6}$$

Whakahekea te hautanga \frac{-62}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{31}{3}$$

Kua oti te whārite te whakatau.

$$x=12 x=-\frac{31}{3}$$

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.

$$3x^{2}-5x-372=0$$

Me tāpiri 372 ki ngā taha e rua o te whārite.

$$3x^{2}-5x-372-\left(-372\right)=-\left(-372\right)$$

Mā te tango i te -372 i a ia ake anō ka toe ko te 0.

$$3x^{2}-5x=-\left(-372\right)$$

Tango -372 mai i 0.

$$3x^{2}-5x=372$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-5x}{3}=\frac{372}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{5}{3}x=\frac{372}{3}$$

Whakawehe 372 ki te 3.

$$x^{2}-\frac{5}{3}x=124$$

Whakawehea te -\frac{5}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{6}. Nā, tāpiria te pūrua o te -\frac{5}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=124+\left(-\frac{5}{6}\right)^{2}$$

Pūruatia -\frac{5}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=124+\frac{25}{36}$$

Tāpiri 124 ki te \frac{25}{36}.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{4489}{36}$$

Tauwehea x^{2}-\frac{5}{3}x+\frac{25}{36}. 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}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{4489}{36}$$

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

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{4489}{36}}$$

Whakarūnātia.

$$x-\frac{5}{6}=\frac{67}{6} x-\frac{5}{6}=-\frac{67}{6}$$

Me tāpiri \frac{5}{6} ki ngā taha e rua o te whārite.

$$x=12 x=-\frac{31}{3}$$