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

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

Whakaoti mō x

$$x=-\frac{5}{9}\approx -0.555555556$$

$$x=0$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+1 ki te 2.

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

Pahekotia te x me 6x, ka 7x.

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

Tāpirihia te 1 ki te 2, ka 3.

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

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

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

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x+3 ki te 3x+1 ka whakakotahi i ngā kupu rite.

$$7x+3=9x^{2}+12x+3$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$7x+3-9x^{2}=12x+3$$

Tangohia te 12x mai i ngā taha e rua.

$$7x+3-9x^{2}-12x=3$$

Pahekotia te 7x me -12x, ka -5x.

$$-5x+3-9x^{2}=3$$

Tangohia te 3 mai i ngā taha e rua.

$$-5x+3-9x^{2}-3=0$$

Tangohia te 3 i te 3, ka 0.

$$-5x-9x^{2}=0$$

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.

$$-9x^{2}-5x=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, -5 mō b, me 0 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te \left(-5\right)^{2}.

$$x=\frac{-\left(-5\right)±5}{2\left(-9\right)}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±5}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{5±5}{-18}$$

Nā, me whakaoti te whārite x=\frac{5±5}{-18} ina he tāpiri te ±. Tāpiri 5 ki te 5.

$$x=\frac{10}{-18}$$

Whakahekea te hautanga \frac{10}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{5}{9}$$

Nā, me whakaoti te whārite x=\frac{5±5}{-18} ina he tango te ±. Tango 5 mai i 5.

$$x=\frac{0}{-18}$$

Whakawehe 0 ki te -18.

$$x=0$$

Kua oti te whārite te whakatau.

$$x=-\frac{5}{9} x=0$$

Ngā Upane mō te Whakaoti i te Pūrua

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+1 ki te 2.

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

Pahekotia te x me 6x, ka 7x.

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

Tāpirihia te 1 ki te 2, ka 3.

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

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

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

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x+3 ki te 3x+1 ka whakakotahi i ngā kupu rite.

$$7x+3=9x^{2}+12x+3$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$7x+3-9x^{2}=12x+3$$

Tangohia te 12x mai i ngā taha e rua.

$$7x+3-9x^{2}-12x=3$$

Pahekotia te 7x me -12x, ka -5x.

$$-5x+3-9x^{2}=3$$

Tangohia te 3 mai i ngā taha e rua.

$$-5x-9x^{2}=3-3$$

Tangohia te 3 i te 3, ka 0.

$$-5x-9x^{2}=0$$

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.

$$-9x^{2}-5x=0$$

Whakawehea ngā taha e rua ki te -9.

$$\frac{-9x^{2}-5x}{-9}=\frac{0}{-9}$$

Mā te whakawehe ki te -9 ka wetekia te whakareanga ki te -9.

$$x^{2}+\left(-\frac{5}{-9}\right)x=\frac{0}{-9}$$

Whakawehe -5 ki te -9.

$$x^{2}+\frac{5}{9}x=\frac{0}{-9}$$

Whakawehe 0 ki te -9.

$$x^{2}+\frac{5}{9}x=0$$

Whakawehea te \frac{5}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{18}. Nā, tāpiria te pūrua o te \frac{5}{18} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{9}x+\left(\frac{5}{18}\right)^{2}=\left(\frac{5}{18}\right)^{2}$$

Pūruatia \frac{5}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=\frac{25}{324}$$

Tauwehea x^{2}+\frac{5}{9}x+\frac{25}{324}. 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}.

$$\left(x+\frac{5}{18}\right)^{2}=\frac{25}{324}$$

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

$$\sqrt{\left(x+\frac{5}{18}\right)^{2}}=\sqrt{\frac{25}{324}}$$

Whakarūnātia.

$$x+\frac{5}{18}=\frac{5}{18} x+\frac{5}{18}=-\frac{5}{18}$$

Me tango \frac{5}{18} mai i ngā taha e rua o te whārite.

$$x=0 x=-\frac{5}{9}$$