Whakaoti i te 3x^2-31x-60=0

$$3 { x }^{ 2 } -31x-60=0$$

Whakaoti mō x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$x=12$$

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-60. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-31 ab=3\left(-60\right)=-180$$

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 -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -31.

$$a=-36 b=5$$

Tuhia anō te 3x^{2}-31x-60 hei \left(3x^{2}-36x\right)+\left(5x-60\right).

$$\left(3x^{2}-36x\right)+\left(5x-60\right)$$

Tauwehea te 3x i te tuatahi me te 5 i te rōpū tuarua.

$$3x\left(x-12\right)+5\left(x-12\right)$$

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

$$\left(x-12\right)\left(3x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te 3x+5=0.

$$x=12 x=-\frac{5}{3}$$

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.

$$3x^{2}-31x-60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -31 mō b, me -60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 3\left(-60\right)}}{2\times 3}$$

Pūrua -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\times 3\left(-60\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-31\right)±\sqrt{961-12\left(-60\right)}}{2\times 3}$$

Whakareatia -12 ki te -60.

$$x=\frac{-\left(-31\right)±\sqrt{961+720}}{2\times 3}$$

Tāpiri 961 ki te 720.

$$x=\frac{-\left(-31\right)±\sqrt{1681}}{2\times 3}$$

Tuhia te pūtakerua o te 1681.

$$x=\frac{-\left(-31\right)±41}{2\times 3}$$

Ko te tauaro o -31 ko 31.

$$x=\frac{31±41}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{31±41}{6}$$

Nā, me whakaoti te whārite x=\frac{31±41}{6} ina he tāpiri te ±. Tāpiri 31 ki te 41.

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

Whakawehe 72 ki te 6.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{31±41}{6} ina he tango te ±. Tango 41 mai i 31.

$$x=-\frac{10}{6}$$

Whakahekea te hautanga \frac{-10}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{5}{3}$$

Kua oti te whārite te whakatau.

$$x=12 x=-\frac{5}{3}$$

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.

$$3x^{2}-31x-60=0$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$3x^{2}-31x-60-\left(-60\right)=-\left(-60\right)$$

Mā te tango i te -60 i a ia ake anō ka toe ko te 0.

$$3x^{2}-31x=-\left(-60\right)$$

Tango -60 mai i 0.

$$3x^{2}-31x=60$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-31x}{3}=\frac{60}{3}$$

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

$$x^{2}-\frac{31}{3}x=\frac{60}{3}$$

Whakawehe 60 ki te 3.

$$x^{2}-\frac{31}{3}x=20$$

Whakawehea te -\frac{31}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{31}{6}. Nā, tāpiria te pūrua o te -\frac{31}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{31}{3}x+\left(-\frac{31}{6}\right)^{2}=20+\left(-\frac{31}{6}\right)^{2}$$

Pūruatia -\frac{31}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=20+\frac{961}{36}$$

Tāpiri 20 ki te \frac{961}{36}.

$$x^{2}-\frac{31}{3}x+\frac{961}{36}=\frac{1681}{36}$$

Tauwehea x^{2}-\frac{31}{3}x+\frac{961}{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}.

$$\left(x-\frac{31}{6}\right)^{2}=\frac{1681}{36}$$

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

$$\sqrt{\left(x-\frac{31}{6}\right)^{2}}=\sqrt{\frac{1681}{36}}$$

Whakarūnātia.

$$x-\frac{31}{6}=\frac{41}{6} x-\frac{31}{6}=-\frac{41}{6}$$

Me tāpiri \frac{31}{6} ki ngā taha e rua o te whārite.

$$x=12 x=-\frac{5}{3}$$