Whakaoti i te 3/2x+1-1/3x+2=2.g

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

Whakaoti mō x

$$x=\frac{\sqrt{97}-7}{24}\approx 0.118702408$$

$$x=\frac{-\sqrt{97}-7}{24}\approx -0.702035742$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{2}{3},-\frac{1}{2} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(2x+1\right)\left(3x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2x+1,3x+2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+2 ki te 3.

$$9x+6-\left(2x+1\right)=2\left(2x+1\right)\left(3x+2\right)$$

Hei kimi i te tauaro o 2x+1, kimihia te tauaro o ia taurangi.

$$9x+6-2x-1=2\left(2x+1\right)\left(3x+2\right)$$

Pahekotia te 9x me -2x, ka 7x.

$$7x+6-1=2\left(2x+1\right)\left(3x+2\right)$$

Tangohia te 1 i te 6, ka 5.

$$7x+5=2\left(2x+1\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 2x+1.

$$7x+5=\left(4x+2\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4x+2 ki te 3x+2 ka whakakotahi i ngā kupu rite.

$$7x+5=12x^{2}+14x+4$$

Tangohia te 12x^{2} mai i ngā taha e rua.

$$7x+5-12x^{2}=14x+4$$

Tangohia te 14x mai i ngā taha e rua.

$$7x+5-12x^{2}-14x=4$$

Pahekotia te 7x me -14x, ka -7x.

$$-7x+5-12x^{2}=4$$

Tangohia te 4 mai i ngā taha e rua.

$$-7x+5-12x^{2}-4=0$$

Tangohia te 4 i te 5, ka 1.

$$-7x+1-12x^{2}=0$$

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.

$$-12x^{2}-7x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -12 mō a, -7 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-12\right)}}{2\left(-12\right)}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-12\right)}}{2\left(-12\right)}$$

Whakareatia -4 ki te -12.

$$x=\frac{-\left(-7\right)±\sqrt{49+48}}{2\left(-12\right)}$$

Tāpiri 49 ki te 48.

$$x=\frac{-\left(-7\right)±\sqrt{97}}{2\left(-12\right)}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{97}}{2\left(-12\right)}$$

Whakareatia 2 ki te -12.

$$x=\frac{7±\sqrt{97}}{-24}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{97}}{-24} ina he tāpiri te ±. Tāpiri 7 ki te \sqrt{97}.

$$x=\frac{\sqrt{97}+7}{-24}$$

Whakawehe 7+\sqrt{97} ki te -24.

$$x=\frac{-\sqrt{97}-7}{24}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{97}}{-24} ina he tango te ±. Tango \sqrt{97} mai i 7.

$$x=\frac{7-\sqrt{97}}{-24}$$

Whakawehe 7-\sqrt{97} ki te -24.

$$x=\frac{\sqrt{97}-7}{24}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{97}-7}{24} x=\frac{\sqrt{97}-7}{24}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{2}{3},-\frac{1}{2} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(2x+1\right)\left(3x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2x+1,3x+2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+2 ki te 3.

$$9x+6-\left(2x+1\right)=2\left(2x+1\right)\left(3x+2\right)$$

Hei kimi i te tauaro o 2x+1, kimihia te tauaro o ia taurangi.

$$9x+6-2x-1=2\left(2x+1\right)\left(3x+2\right)$$

Pahekotia te 9x me -2x, ka 7x.

$$7x+6-1=2\left(2x+1\right)\left(3x+2\right)$$

Tangohia te 1 i te 6, ka 5.

$$7x+5=2\left(2x+1\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 2x+1.

$$7x+5=\left(4x+2\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4x+2 ki te 3x+2 ka whakakotahi i ngā kupu rite.

$$7x+5=12x^{2}+14x+4$$

Tangohia te 12x^{2} mai i ngā taha e rua.

$$7x+5-12x^{2}=14x+4$$

Tangohia te 14x mai i ngā taha e rua.

$$7x+5-12x^{2}-14x=4$$

Pahekotia te 7x me -14x, ka -7x.

$$-7x+5-12x^{2}=4$$

Tangohia te 5 mai i ngā taha e rua.

$$-7x-12x^{2}=4-5$$

Tangohia te 5 i te 4, ka -1.

$$-7x-12x^{2}=-1$$

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.

$$-12x^{2}-7x=-1$$

Whakawehea ngā taha e rua ki te -12.

$$\frac{-12x^{2}-7x}{-12}=-\frac{1}{-12}$$

Mā te whakawehe ki te -12 ka wetekia te whakareanga ki te -12.

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

Whakawehe -7 ki te -12.

$$x^{2}+\frac{7}{12}x=-\frac{1}{-12}$$

Whakawehe -1 ki te -12.

$$x^{2}+\frac{7}{12}x=\frac{1}{12}$$

Whakawehea te \frac{7}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{24}. Nā, tāpiria te pūrua o te \frac{7}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{12}x+\left(\frac{7}{24}\right)^{2}=\frac{1}{12}+\left(\frac{7}{24}\right)^{2}$$

Pūruatia \frac{7}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{12}x+\frac{49}{576}=\frac{1}{12}+\frac{49}{576}$$

Tāpiri \frac{1}{12} ki te \frac{49}{576} 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.

$$x^{2}+\frac{7}{12}x+\frac{49}{576}=\frac{97}{576}$$

Tauwehea x^{2}+\frac{7}{12}x+\frac{49}{576}. 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}.

$$\left(x+\frac{7}{24}\right)^{2}=\frac{97}{576}$$

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

$$\sqrt{\left(x+\frac{7}{24}\right)^{2}}=\sqrt{\frac{97}{576}}$$

Whakarūnātia.

$$x+\frac{7}{24}=\frac{\sqrt{97}}{24} x+\frac{7}{24}=-\frac{\sqrt{97}}{24}$$

Me tango \frac{7}{24} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{97}-7}{24} x=\frac{-\sqrt{97}-7}{24}$$