Whakaoti i te x+3x+1/2-x-2/3=x^2-2

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

Whakaoti mō x

$$x=-1$$

$$x = \frac{19}{6} = 3\frac{1}{6} \approx 3.166666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,3.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x+1.

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

Pahekotia te 6x me 9x, ka 15x.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-2.

$$15x+3-2x+4=6x^{2}-12$$

Pahekotia te 15x me -2x, ka 13x.

$$13x+3+4=6x^{2}-12$$

Tāpirihia te 3 ki te 4, ka 7.

$$13x+7=6x^{2}-12$$

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

$$13x+7-6x^{2}=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$13x+7-6x^{2}+12=0$$

Tāpirihia te 7 ki te 12, ka 19.

$$13x+19-6x^{2}=0$$

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

$$-6x^{2}+13x+19=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -6x^{2}+ax+bx+19. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=-6\times 19=-114$$

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 -114.

$$-1,114 -2,57 -3,38 -6,19$$

Tātaihia te tapeke mō ia takirua.

$$-1+114=113 -2+57=55 -3+38=35 -6+19=13$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=19 b=-6$$

Tuhia anō te -6x^{2}+13x+19 hei \left(-6x^{2}+19x\right)+\left(-6x+19\right).

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

Tauwehea te -x i te tuatahi me te -1 i te rōpū tuarua.

$$-x\left(6x-19\right)-\left(6x-19\right)$$

Whakatauwehea atu te kīanga pātahi 6x-19 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6x-19\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 6x-19=0 me te -x-1=0.

$$x=\frac{19}{6} x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,3.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x+1.

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

Pahekotia te 6x me 9x, ka 15x.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-2.

$$15x+3-2x+4=6x^{2}-12$$

Pahekotia te 15x me -2x, ka 13x.

$$13x+3+4=6x^{2}-12$$

Tāpirihia te 3 ki te 4, ka 7.

$$13x+7=6x^{2}-12$$

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

$$13x+7-6x^{2}=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$13x+7-6x^{2}+12=0$$

Tāpirihia te 7 ki te 12, ka 19.

$$13x+19-6x^{2}=0$$

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.

$$-6x^{2}+13x+19=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, 13 mō b, me 19 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-13±\sqrt{169+24\times 19}}{2\left(-6\right)}$$

Whakareatia 24 ki te 19.

$$x=\frac{-13±\sqrt{169+456}}{2\left(-6\right)}$$

Tāpiri 169 ki te 456.

$$x=\frac{-13±\sqrt{625}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te 625.

$$x=\frac{-13±25}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{-13±25}{-12}$$

Nā, me whakaoti te whārite x=\frac{-13±25}{-12} ina he tāpiri te ±. Tāpiri -13 ki te 25.

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

Whakawehe 12 ki te -12.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-13±25}{-12} ina he tango te ±. Tango 25 mai i -13.

$$x=-\frac{38}{-12}$$

Whakahekea te hautanga \frac{-38}{-12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{19}{6}$$

Kua oti te whārite te whakatau.

$$x=-1 x=\frac{19}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 2,3.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 3x+1.

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

Pahekotia te 6x me 9x, ka 15x.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

Whakamahia te āhuatanga tohatoha hei whakarea te -2 ki te x-2.

$$15x+3-2x+4=6x^{2}-12$$

Pahekotia te 15x me -2x, ka 13x.

$$13x+3+4=6x^{2}-12$$

Tāpirihia te 3 ki te 4, ka 7.

$$13x+7=6x^{2}-12$$

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

$$13x+7-6x^{2}=-12$$

Tangohia te 7 mai i ngā taha e rua.

$$13x-6x^{2}=-12-7$$

Tangohia te 7 i te -12, ka -19.

$$13x-6x^{2}=-19$$

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.

$$-6x^{2}+13x=-19$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}+13x}{-6}=-\frac{19}{-6}$$

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

$$x^{2}+\frac{13}{-6}x=-\frac{19}{-6}$$

Whakawehe 13 ki te -6.

$$x^{2}-\frac{13}{6}x=-\frac{19}{-6}$$

Whakawehe -19 ki te -6.

$$x^{2}-\frac{13}{6}x=\frac{19}{6}$$

Whakawehea te -\frac{13}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{12}. Nā, tāpiria te pūrua o te -\frac{13}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=\frac{19}{6}+\left(-\frac{13}{12}\right)^{2}$$

Pūruatia -\frac{13}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{19}{6}+\frac{169}{144}$$

Tāpiri \frac{19}{6} ki te \frac{169}{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.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{625}{144}$$

Tauwehea te x^{2}-\frac{13}{6}x+\frac{169}{144}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{625}{144}$$

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

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{625}{144}}$$

Whakarūnātia.

$$x-\frac{13}{12}=\frac{25}{12} x-\frac{13}{12}=-\frac{25}{12}$$

Me tāpiri \frac{13}{12} ki ngā taha e rua o te whārite.

$$x=\frac{19}{6} x=-1$$