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

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

Whakaoti mō x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=-\frac{1}{2}=-0.5$$

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+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=6\times 2=12$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 12.

$$1,12 2,6 3,4$$

Tātaihia te tapeke mō ia takirua.

$$1+12=13 2+6=8 3+4=7$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=3 b=4$$

Tuhia anō te 6x^{2}+7x+2 hei \left(6x^{2}+3x\right)+\left(4x+2\right).

$$\left(6x^{2}+3x\right)+\left(4x+2\right)$$

Tauwehea te 3x i te tuatahi me te 2 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 2x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

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

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

$$x=\frac{-7±\sqrt{7^{2}-4\times 6\times 2}}{2\times 6}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 6\times 2}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-7±\sqrt{49-24\times 2}}{2\times 6}$$

Whakareatia -24 ki te 2.

$$x=\frac{-7±\sqrt{49-48}}{2\times 6}$$

Tāpiri 49 ki te -48.

$$x=\frac{-7±\sqrt{1}}{2\times 6}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-7±1}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-7±1}{12}$$

Nā, me whakaoti te whārite x=\frac{-7±1}{12} ina he tāpiri te ±. Tāpiri -7 ki te 1.

$$x=-\frac{6}{12}$$

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

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

Nā, me whakaoti te whārite x=\frac{-7±1}{12} ina he tango te ±. Tango 1 mai i -7.

$$x=-\frac{8}{12}$$

Whakahekea te hautanga \frac{-8}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

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

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.

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

Me tango 2 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te 6.

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

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

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

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

$$x^{2}+\frac{7}{6}x=-\frac{1}{3}$$

Whakawehea te \frac{7}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{12}. Nā, tāpiria te pūrua o te \frac{7}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{6}x+\left(\frac{7}{12}\right)^{2}=-\frac{1}{3}+\left(\frac{7}{12}\right)^{2}$$

Pūruatia \frac{7}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=-\frac{1}{3}+\frac{49}{144}$$

Tāpiri -\frac{1}{3} ki te \frac{49}{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.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{1}{144}$$

Tauwehea x^{2}+\frac{7}{6}x+\frac{49}{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}.

$$\left(x+\frac{7}{12}\right)^{2}=\frac{1}{144}$$

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

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Whakarūnātia.

$$x+\frac{7}{12}=\frac{1}{12} x+\frac{7}{12}=-\frac{1}{12}$$

Me tango \frac{7}{12} mai i ngā taha e rua o te whārite.

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