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

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

Whakaoti mō u

$$u=-\frac{1}{5}=-0.2$$

$$u=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3 mai i ngā taha e rua.

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

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

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

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

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

Whakatauwehea atu 5u i te 5u^{2}-15u.

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

Whakatauwehea atu te kīanga pātahi u-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

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

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

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

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

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

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

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

Pūrua -14.

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

Whakareatia -4 ki te 5.

$$u=\frac{-\left(-14\right)±\sqrt{196-20\left(-3\right)}}{2\times 5}$$

Whakareatia -20 ki te -3.

$$u=\frac{-\left(-14\right)±\sqrt{196+60}}{2\times 5}$$

Tāpiri 196 ki te 60.

$$u=\frac{-\left(-14\right)±\sqrt{256}}{2\times 5}$$

Tuhia te pūtakerua o te 256.

$$u=\frac{-\left(-14\right)±16}{2\times 5}$$

Ko te tauaro o -14 ko 14.

$$u=\frac{14±16}{2\times 5}$$

Whakareatia 2 ki te 5.

$$u=\frac{14±16}{10}$$

Nā, me whakaoti te whārite u=\frac{14±16}{10} ina he tāpiri te ±. Tāpiri 14 ki te 16.

$$u=\frac{30}{10}$$

Whakawehe 30 ki te 10.

$$u=3$$

Nā, me whakaoti te whārite u=\frac{14±16}{10} ina he tango te ±. Tango 16 mai i 14.

$$u=-\frac{2}{10}$$

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

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

Kua oti te whārite te whakatau.

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

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.

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

Whakawehea ngā taha e rua ki te 5.

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

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

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

Whakawehea te -\frac{14}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{5}. Nā, tāpiria te pūrua o te -\frac{7}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{7}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$u^{2}-\frac{14}{5}u+\frac{49}{25}=\frac{3}{5}+\frac{49}{25}$$

Tāpiri \frac{3}{5} ki te \frac{49}{25} 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.

$$u^{2}-\frac{14}{5}u+\frac{49}{25}=\frac{64}{25}$$

Tauwehea u^{2}-\frac{14}{5}u+\frac{49}{25}. 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}.

$$\left(u-\frac{7}{5}\right)^{2}=\frac{64}{25}$$

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

$$\sqrt{\left(u-\frac{7}{5}\right)^{2}}=\sqrt{\frac{64}{25}}$$

Whakarūnātia.

$$u-\frac{7}{5}=\frac{8}{5} u-\frac{7}{5}=-\frac{8}{5}$$

Me tāpiri \frac{7}{5} ki ngā taha e rua o te whārite.

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