Whakaoti i te -9x=2(3x^2+4)+10x

$$- 9 x = 2 ( 3 x ^ { 2 } + 4 ) + 10 x$$

Whakaoti mō x

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

$$x = -\frac{8}{3} = -2\frac{2}{3} \approx -2.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x^{2}+4.

$$-9x=6x^{2}+8+10x$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$-9x-6x^{2}=8+10x$$

Tangohia te 8 mai i ngā taha e rua.

$$-9x-6x^{2}-8=10x$$

Tangohia te 10x mai i ngā taha e rua.

$$-9x-6x^{2}-8-10x=0$$

Pahekotia te -9x me -10x, ka -19x.

$$-19x-6x^{2}-8=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-6x^{2}-19x-8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -6x^{2}+ax+bx-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-19 ab=-6\left(-8\right)=48$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

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

$$a=-3 b=-16$$

Tuhia anō te -6x^{2}-19x-8 hei \left(-6x^{2}-3x\right)+\left(-16x-8\right).

$$\left(-6x^{2}-3x\right)+\left(-16x-8\right)$$

Tauwehea te -3x i te tuatahi me te -8 i te rōpū tuarua.

$$-3x\left(2x+1\right)-8\left(2x+1\right)$$

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

$$\left(2x+1\right)\left(-3x-8\right)$$

Hei kimi otinga whārite, me whakaoti te 2x+1=0 me te -3x-8=0.

$$x=-\frac{1}{2} x=-\frac{8}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x^{2}+4.

$$-9x=6x^{2}+8+10x$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$-9x-6x^{2}=8+10x$$

Tangohia te 8 mai i ngā taha e rua.

$$-9x-6x^{2}-8=10x$$

Tangohia te 10x mai i ngā taha e rua.

$$-9x-6x^{2}-8-10x=0$$

Pahekotia te -9x me -10x, ka -19x.

$$-19x-6x^{2}-8=0$$

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.

$$-6x^{2}-19x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, -19 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\left(-6\right)\left(-8\right)}}{2\left(-6\right)}$$

Pūrua -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\left(-6\right)\left(-8\right)}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-\left(-19\right)±\sqrt{361+24\left(-8\right)}}{2\left(-6\right)}$$

Whakareatia 24 ki te -8.

$$x=\frac{-\left(-19\right)±\sqrt{361-192}}{2\left(-6\right)}$$

Tāpiri 361 ki te -192.

$$x=\frac{-\left(-19\right)±\sqrt{169}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-19\right)±13}{2\left(-6\right)}$$

Ko te tauaro o -19 ko 19.

$$x=\frac{19±13}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{19±13}{-12}$$

Nā, me whakaoti te whārite x=\frac{19±13}{-12} ina he tāpiri te ±. Tāpiri 19 ki te 13.

$$x=\frac{32}{-12}$$

Whakahekea te hautanga \frac{32}{-12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{8}{3}$$

Nā, me whakaoti te whārite x=\frac{19±13}{-12} ina he tango te ±. Tango 13 mai i 19.

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

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

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

Kua oti te whārite te whakatau.

$$x=-\frac{8}{3} x=-\frac{1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x^{2}+4.

$$-9x=6x^{2}+8+10x$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$-9x-6x^{2}=8+10x$$

Tangohia te 10x mai i ngā taha e rua.

$$-9x-6x^{2}-10x=8$$

Pahekotia te -9x me -10x, ka -19x.

$$-19x-6x^{2}=8$$

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.

$$-6x^{2}-19x=8$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}-19x}{-6}=\frac{8}{-6}$$

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

$$x^{2}+\left(-\frac{19}{-6}\right)x=\frac{8}{-6}$$

Whakawehe -19 ki te -6.

$$x^{2}+\frac{19}{6}x=\frac{8}{-6}$$

Whakahekea te hautanga \frac{8}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{19}{6}x=-\frac{4}{3}$$

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.

$$x^{2}+\frac{19}{6}x+\left(\frac{19}{12}\right)^{2}=-\frac{4}{3}+\left(\frac{19}{12}\right)^{2}$$

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

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=-\frac{4}{3}+\frac{361}{144}$$

Tāpiri -\frac{4}{3} 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.

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=\frac{169}{144}$$

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}.

$$\left(x+\frac{19}{12}\right)^{2}=\frac{169}{144}$$

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

$$\sqrt{\left(x+\frac{19}{12}\right)^{2}}=\sqrt{\frac{169}{144}}$$

Whakarūnātia.

$$x+\frac{19}{12}=\frac{13}{12} x+\frac{19}{12}=-\frac{13}{12}$$

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

$$x=-\frac{1}{2} x=-\frac{8}{3}$$