Whakaoti i te 3u^2-14u=5

$$3 u ^ { 2 } - 14 u = 5$$

Whakaoti mō u

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

$$u=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 5 mai i ngā taha e rua.

$$3u^{2}-14u-5=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3u^{2}+au+bu-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=3\left(-5\right)=-15$$

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 -15.

$$1,-15 3,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-15=-14 3-5=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-15 b=1$$

Tuhia anō te 3u^{2}-14u-5 hei \left(3u^{2}-15u\right)+\left(u-5\right).

$$\left(3u^{2}-15u\right)+\left(u-5\right)$$

Whakatauwehea atu 3u i te 3u^{2}-15u.

$$3u\left(u-5\right)+u-5$$

Whakatauwehea atu te kīanga pātahi u-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(u-5\right)\left(3u+1\right)$$

Hei kimi otinga whārite, me whakaoti te u-5=0 me te 3u+1=0.

$$u=5 u=-\frac{1}{3}$$

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.

$$3u^{2}-14u=5$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$3u^{2}-14u-5=5-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$3u^{2}-14u-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -14 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 3\left(-5\right)}}{2\times 3}$$

Pūrua -14.

$$u=\frac{-\left(-14\right)±\sqrt{196-4\times 3\left(-5\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$u=\frac{-\left(-14\right)±\sqrt{196-12\left(-5\right)}}{2\times 3}$$

Whakareatia -12 ki te -5.

$$u=\frac{-\left(-14\right)±\sqrt{196+60}}{2\times 3}$$

Tāpiri 196 ki te 60.

$$u=\frac{-\left(-14\right)±\sqrt{256}}{2\times 3}$$

Tuhia te pūtakerua o te 256.

$$u=\frac{-\left(-14\right)±16}{2\times 3}$$

Ko te tauaro o -14 ko 14.

$$u=\frac{14±16}{2\times 3}$$

Whakareatia 2 ki te 3.

$$u=\frac{14±16}{6}$$

Nā, me whakaoti te whārite u=\frac{14±16}{6} ina he tāpiri te ±. Tāpiri 14 ki te 16.

$$u=\frac{30}{6}$$

Whakawehe 30 ki te 6.

$$u=5$$

Nā, me whakaoti te whārite u=\frac{14±16}{6} ina he tango te ±. Tango 16 mai i 14.

$$u=-\frac{2}{6}$$

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

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

Kua oti te whārite te whakatau.

$$u=5 u=-\frac{1}{3}$$

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.

$$3u^{2}-14u=5$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3u^{2}-14u}{3}=\frac{5}{3}$$

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

$$u^{2}-\frac{14}{3}u=\frac{5}{3}$$

Whakawehea te -\frac{14}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{3}. Nā, tāpiria te pūrua o te -\frac{7}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$u^{2}-\frac{14}{3}u+\left(-\frac{7}{3}\right)^{2}=\frac{5}{3}+\left(-\frac{7}{3}\right)^{2}$$

Pūruatia -\frac{7}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$u^{2}-\frac{14}{3}u+\frac{49}{9}=\frac{5}{3}+\frac{49}{9}$$

Tāpiri \frac{5}{3} ki te \frac{49}{9} 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.

$$u^{2}-\frac{14}{3}u+\frac{49}{9}=\frac{64}{9}$$

Tauwehea u^{2}-\frac{14}{3}u+\frac{49}{9}. 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}.

$$\left(u-\frac{7}{3}\right)^{2}=\frac{64}{9}$$

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

$$\sqrt{\left(u-\frac{7}{3}\right)^{2}}=\sqrt{\frac{64}{9}}$$

Whakarūnātia.

$$u-\frac{7}{3}=\frac{8}{3} u-\frac{7}{3}=-\frac{8}{3}$$

Me tāpiri \frac{7}{3} ki ngā taha e rua o te whārite.

$$u=5 u=-\frac{1}{3}$$