Whakaoti i te 4/x+3+3/2x-1=1

$$\frac { 4 } { x + 3 } + \frac { 3 } { 2 x - 1 } = 1$$

Whakaoti mō x

$$x=-1$$

$$x=4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -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(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+3,2x-1.

$$\left(2x-1\right)\times 4+\left(x+3\right)\times 3=\left(2x-1\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-1 ki te 4.

$$8x-4+\left(x+3\right)\times 3=\left(2x-1\right)\left(x+3\right)$$

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

$$8x-4+3x+9=\left(2x-1\right)\left(x+3\right)$$

Pahekotia te 8x me 3x, ka 11x.

$$11x-4+9=\left(2x-1\right)\left(x+3\right)$$

Tāpirihia te -4 ki te 9, ka 5.

$$11x+5=\left(2x-1\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-1 ki te x+3 ka whakakotahi i ngā kupu rite.

$$11x+5=2x^{2}+5x-3$$

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

$$11x+5-2x^{2}=5x-3$$

Tangohia te 5x mai i ngā taha e rua.

$$11x+5-2x^{2}-5x=-3$$

Pahekotia te 11x me -5x, ka 6x.

$$6x+5-2x^{2}=-3$$

Me tāpiri te 3 ki ngā taha e rua.

$$6x+5-2x^{2}+3=0$$

Tāpirihia te 5 ki te 3, ka 8.

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

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.

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

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

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

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-2\right)\times 8}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-6±\sqrt{36+8\times 8}}{2\left(-2\right)}$$

Whakareatia 8 ki te 8.

$$x=\frac{-6±\sqrt{36+64}}{2\left(-2\right)}$$

Tāpiri 36 ki te 64.

$$x=\frac{-6±\sqrt{100}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-6±10}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-6±10}{-4}$$

Nā, me whakaoti te whārite x=\frac{-6±10}{-4} ina he tāpiri te ±. Tāpiri -6 ki te 10.

$$x=\frac{4}{-4}$$

Whakawehe 4 ki te -4.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-6±10}{-4} ina he tango te ±. Tango 10 mai i -6.

$$x=-\frac{16}{-4}$$

Whakawehe -16 ki te -4.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=-1 x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -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(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+3,2x-1.

$$\left(2x-1\right)\times 4+\left(x+3\right)\times 3=\left(2x-1\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x-1 ki te 4.

$$8x-4+\left(x+3\right)\times 3=\left(2x-1\right)\left(x+3\right)$$

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

$$8x-4+3x+9=\left(2x-1\right)\left(x+3\right)$$

Pahekotia te 8x me 3x, ka 11x.

$$11x-4+9=\left(2x-1\right)\left(x+3\right)$$

Tāpirihia te -4 ki te 9, ka 5.

$$11x+5=\left(2x-1\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-1 ki te x+3 ka whakakotahi i ngā kupu rite.

$$11x+5=2x^{2}+5x-3$$

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

$$11x+5-2x^{2}=5x-3$$

Tangohia te 5x mai i ngā taha e rua.

$$11x+5-2x^{2}-5x=-3$$

Pahekotia te 11x me -5x, ka 6x.

$$6x+5-2x^{2}=-3$$

Tangohia te 5 mai i ngā taha e rua.

$$6x-2x^{2}=-3-5$$

Tangohia te 5 i te -3, ka -8.

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

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.

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

Whakawehea ngā taha e rua ki te -2.

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

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

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

Whakawehe 6 ki te -2.

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

Whakawehe -8 ki te -2.

$$x^{2}-3x=4$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=4+\frac{9}{4}$$

Tāpiri 4 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{25}{4}$$

Tauwehea x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{5}{2} x-\frac{3}{2}=-\frac{5}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=4 x=-1$$