Whakaoti i te 3x^2+5x=138

$$3 { x }^{ 2 } +5x=138$$

Whakaoti mō x

$$x = -\frac{23}{3} = -7\frac{2}{3} \approx -7.666666667$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 138 mai i ngā taha e rua.

$$3x^{2}+5x-138=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-138. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=3\left(-138\right)=-414$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -414.

$$-1,414 -2,207 -3,138 -6,69 -9,46 -18,23$$

Tātaihia te tapeke mō ia takirua.

$$-1+414=413 -2+207=205 -3+138=135 -6+69=63 -9+46=37 -18+23=5$$

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

$$a=-18 b=23$$

Tuhia anō te 3x^{2}+5x-138 hei \left(3x^{2}-18x\right)+\left(23x-138\right).

$$\left(3x^{2}-18x\right)+\left(23x-138\right)$$

Tauwehea te 3x i te tuatahi me te 23 i te rōpū tuarua.

$$3x\left(x-6\right)+23\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(3x+23\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te 3x+23=0.

$$x=6 x=-\frac{23}{3}$$

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.

$$3x^{2}+5x=138$$

Me tango 138 mai i ngā taha e rua o te whārite.

$$3x^{2}+5x-138=138-138$$

Mā te tango i te 138 i a ia ake anō ka toe ko te 0.

$$3x^{2}+5x-138=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 5 mō b, me -138 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\left(-138\right)}}{2\times 3}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 3\left(-138\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-5±\sqrt{25-12\left(-138\right)}}{2\times 3}$$

Whakareatia -12 ki te -138.

$$x=\frac{-5±\sqrt{25+1656}}{2\times 3}$$

Tāpiri 25 ki te 1656.

$$x=\frac{-5±\sqrt{1681}}{2\times 3}$$

Tuhia te pūtakerua o te 1681.

$$x=\frac{-5±41}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-5±41}{6}$$

Nā, me whakaoti te whārite x=\frac{-5±41}{6} ina he tāpiri te ±. Tāpiri -5 ki te 41.

$$x=\frac{36}{6}$$

Whakawehe 36 ki te 6.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{-5±41}{6} ina he tango te ±. Tango 41 mai i -5.

$$x=-\frac{46}{6}$$

Whakahekea te hautanga \frac{-46}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{23}{3}$$

Kua oti te whārite te whakatau.

$$x=6 x=-\frac{23}{3}$$

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.

$$3x^{2}+5x=138$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+5x}{3}=\frac{138}{3}$$

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

$$x^{2}+\frac{5}{3}x=\frac{138}{3}$$

Whakawehe 138 ki te 3.

$$x^{2}+\frac{5}{3}x=46$$

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.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=46+\left(\frac{5}{6}\right)^{2}$$

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

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=46+\frac{25}{36}$$

Tāpiri 46 ki te \frac{25}{36}.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{1681}{36}$$

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}.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{1681}{36}$$

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

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{1681}{36}}$$

Whakarūnātia.

$$x+\frac{5}{6}=\frac{41}{6} x+\frac{5}{6}=-\frac{41}{6}$$

Me tango \frac{5}{6} mai i ngā taha e rua o te whārite.

$$x=6 x=-\frac{23}{3}$$