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

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

Whakaoti mō x

$$x=4$$

$$x=0$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$\left(x-2\right)\left(x-1\right)\left(x+3\right)-\left(x^{2}-4\right)\left(2x-3\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

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

$$\left(x^{2}-3x+2\right)\left(x+3\right)-\left(x^{2}-4\right)\left(2x-3\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

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

$$x^{3}-7x+6-\left(x^{2}-4\right)\left(2x-3\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

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

$$x^{3}-7x+6-\left(2x^{3}-3x^{2}-8x+12\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

Hei kimi i te tauaro o 2x^{3}-3x^{2}-8x+12, kimihia te tauaro o ia taurangi.

$$x^{3}-7x+6-2x^{3}+3x^{2}+8x-12=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

Pahekotia te x^{3} me -2x^{3}, ka -x^{3}.

$$-x^{3}-7x+6+3x^{2}+8x-12=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

Pahekotia te -7x me 8x, ka x.

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

Tangohia te 12 i te 6, ka -6.

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

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

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

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

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

Whakamahia te āhuatanga tuaritanga hei whakarea te 2-x-x^{2} ki te x-3 ka whakakotahi i ngā kupu rite.

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

Tangohia te 5x mai i ngā taha e rua.

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

Pahekotia te x me -5x, ka -4x.

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

Tangohia te -6 mai i ngā taha e rua.

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

Ko te tauaro o -6 ko 6.

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

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

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

Tāpirihia te -6 ki te 6, ka 0.

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

Pahekotia te 3x^{2} me -2x^{2}, ka x^{2}.

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

Me tāpiri te x^{3} ki ngā taha e rua.

$$-x^{3}-4x+x^{2}+x^{3}=0$$

Pahekotia te -x^{3} me x^{3}, ka 0.

$$-4x+x^{2}=0$$

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.

$$x^{2}-4x=0$$

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

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

Tuhia te pūtakerua o te \left(-4\right)^{2}.

$$x=\frac{-\left(-4\right)±4}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±4}{2}$$

Nā, me whakaoti te whārite x=\frac{4±4}{2} ina he tāpiri te ±. Tāpiri 4 ki te 4.

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{4±4}{2} ina he tango te ±. Tango 4 mai i 4.

$$x=\frac{0}{2}$$

Whakawehe 0 ki te 2.

$$x=0$$

Kua oti te whārite te whakatau.

$$x=4 x=0$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$\left(x-2\right)\left(x-1\right)\left(x+3\right)-\left(x^{2}-4\right)\left(2x-3\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

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

$$\left(x^{2}-3x+2\right)\left(x+3\right)-\left(x^{2}-4\right)\left(2x-3\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

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

$$x^{3}-7x+6-\left(x^{2}-4\right)\left(2x-3\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

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

$$x^{3}-7x+6-\left(2x^{3}-3x^{2}-8x+12\right)=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

Hei kimi i te tauaro o 2x^{3}-3x^{2}-8x+12, kimihia te tauaro o ia taurangi.

$$x^{3}-7x+6-2x^{3}+3x^{2}+8x-12=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

Pahekotia te x^{3} me -2x^{3}, ka -x^{3}.

$$-x^{3}-7x+6+3x^{2}+8x-12=-\left(-1+x\right)\left(2+x\right)\left(x-3\right)$$

Pahekotia te -7x me 8x, ka x.

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

Tangohia te 12 i te 6, ka -6.

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

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

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

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

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

Whakamahia te āhuatanga tuaritanga hei whakarea te 2-x-x^{2} ki te x-3 ka whakakotahi i ngā kupu rite.

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

Tangohia te 5x mai i ngā taha e rua.

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

Pahekotia te x me -5x, ka -4x.

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

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

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

Pahekotia te 3x^{2} me -2x^{2}, ka x^{2}.

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

Me tāpiri te x^{3} ki ngā taha e rua.

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

Pahekotia te -x^{3} me x^{3}, ka 0.

$$-4x-6+x^{2}=-6$$

Me tāpiri te 6 ki ngā taha e rua.

$$-4x+x^{2}=-6+6$$

Tāpirihia te -6 ki te 6, ka 0.

$$-4x+x^{2}=0$$

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.

$$x^{2}-4x=0$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūrua -2.

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

Tauwehea x^{2}-4x+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}.

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

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

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

Whakarūnātia.

$$x-2=2 x-2=-2$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=4 x=0$$