Whakaoti i te 6x^2+19x-7=0

$$6 { x }^{ 2 } +19x-7=0$$

Whakaoti mō x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=\frac{1}{3}\approx 0.333333333$$

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 6x^{2}+ax+bx-7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=19 ab=6\left(-7\right)=-42$$

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 -42.

$$-1,42 -2,21 -3,14 -6,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

Ko te otinga te takirua ka hoatu i te tapeke 19.

$$a=-2 b=21$$

Tuhia anō te 6x^{2}+19x-7 hei \left(6x^{2}-2x\right)+\left(21x-7\right).

$$\left(6x^{2}-2x\right)+\left(21x-7\right)$$

Tauwehea te 2x i te tuatahi me te 7 i te rōpū tuarua.

$$2x\left(3x-1\right)+7\left(3x-1\right)$$

Whakatauwehea atu te kīanga pātahi 3x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-1\right)\left(2x+7\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-1=0 me te 2x+7=0.

$$x=\frac{1}{3} x=-\frac{7}{2}$$

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.

$$6x^{2}+19x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 19 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\times 6\left(-7\right)}}{2\times 6}$$

Pūrua 19.

$$x=\frac{-19±\sqrt{361-4\times 6\left(-7\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-19±\sqrt{361-24\left(-7\right)}}{2\times 6}$$

Whakareatia -24 ki te -7.

$$x=\frac{-19±\sqrt{361+168}}{2\times 6}$$

Tāpiri 361 ki te 168.

$$x=\frac{-19±\sqrt{529}}{2\times 6}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-19±23}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-19±23}{12}$$

Nā, me whakaoti te whārite x=\frac{-19±23}{12} ina he tāpiri te ±. Tāpiri -19 ki te 23.

$$x=\frac{4}{12}$$

Whakahekea te hautanga \frac{4}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{1}{3}$$

Nā, me whakaoti te whārite x=\frac{-19±23}{12} ina he tango te ±. Tango 23 mai i -19.

$$x=-\frac{42}{12}$$

Whakahekea te hautanga \frac{-42}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{7}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{3} x=-\frac{7}{2}$$

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.

$$6x^{2}+19x-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$6x^{2}+19x-7-\left(-7\right)=-\left(-7\right)$$

Mā te tango i te -7 i a ia ake anō ka toe ko te 0.

$$6x^{2}+19x=-\left(-7\right)$$

Tango -7 mai i 0.

$$6x^{2}+19x=7$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+19x}{6}=\frac{7}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{19}{6}x=\frac{7}{6}$$

Whakawehea te \frac{19}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{19}{12}. Nā, tāpiria te pūrua o te \frac{19}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{19}{6}x+\left(\frac{19}{12}\right)^{2}=\frac{7}{6}+\left(\frac{19}{12}\right)^{2}$$

Pūruatia \frac{19}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=\frac{7}{6}+\frac{361}{144}$$

Tāpiri \frac{7}{6} ki te \frac{361}{144} 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.

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=\frac{529}{144}$$

Tauwehea x^{2}+\frac{19}{6}x+\frac{361}{144}. 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}.

$$\left(x+\frac{19}{12}\right)^{2}=\frac{529}{144}$$

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

$$\sqrt{\left(x+\frac{19}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Whakarūnātia.

$$x+\frac{19}{12}=\frac{23}{12} x+\frac{19}{12}=-\frac{23}{12}$$

Me tango \frac{19}{12} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{3} x=-\frac{7}{2}$$