Whakaoti i te 3/x-3+x+1/x^2-5x+6=1

$$\frac { 3 } { x - 3 } + \frac { x + 1 } { x ^ { 2 } - 5 x + 6 } = 1$$

Whakaoti mō x

$$x = \frac{\sqrt{37} + 9}{2} \approx 7.541381265$$

$$x = \frac{9 - \sqrt{37}}{2} \approx 1.458618735$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 2,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x-2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x^{2}-5x+6.

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

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

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

Pahekotia te 3x me x, ka 4x.

$$4x-6+1=\left(x-3\right)\left(x-2\right)$$

Tāpirihia te -6 ki te 1, ka -5.

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

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

$$4x-5=x^{2}-5x+6$$

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

$$4x-5-x^{2}=-5x+6$$

Me tāpiri te 5x ki ngā taha e rua.

$$4x-5-x^{2}+5x=6$$

Pahekotia te 4x me 5x, ka 9x.

$$9x-5-x^{2}=6$$

Tangohia te 6 mai i ngā taha e rua.

$$9x-5-x^{2}-6=0$$

Tangohia te 6 i te -5, ka -11.

$$9x-11-x^{2}=0$$

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.

$$-x^{2}+9x-11=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 9 mō b, me -11 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-11\right)}}{2\left(-1\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-11\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-9±\sqrt{81+4\left(-11\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -11.

$$x=\frac{-9±\sqrt{81-44}}{2\left(-1\right)}$$

Tāpiri 81 ki te -44.

$$x=\frac{-9±\sqrt{37}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-9±\sqrt{37}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{37}}{-2} ina he tāpiri te ±. Tāpiri -9 ki te \sqrt{37}.

$$x=\frac{\sqrt{37}-9}{-2}$$

Whakawehe -9+\sqrt{37} ki te -2.

$$x=\frac{9-\sqrt{37}}{2}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{37}}{-2} ina he tango te ±. Tango \sqrt{37} mai i -9.

$$x=\frac{-\sqrt{37}-9}{-2}$$

Whakawehe -9-\sqrt{37} ki te -2.

$$x=\frac{\sqrt{37}+9}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{9-\sqrt{37}}{2} x=\frac{\sqrt{37}+9}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 2,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x-2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x^{2}-5x+6.

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

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

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

Pahekotia te 3x me x, ka 4x.

$$4x-6+1=\left(x-3\right)\left(x-2\right)$$

Tāpirihia te -6 ki te 1, ka -5.

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

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

$$4x-5=x^{2}-5x+6$$

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

$$4x-5-x^{2}=-5x+6$$

Me tāpiri te 5x ki ngā taha e rua.

$$4x-5-x^{2}+5x=6$$

Pahekotia te 4x me 5x, ka 9x.

$$9x-5-x^{2}=6$$

Me tāpiri te 5 ki ngā taha e rua.

$$9x-x^{2}=6+5$$

Tāpirihia te 6 ki te 5, ka 11.

$$9x-x^{2}=11$$

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.

$$-x^{2}+9x=11$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+9x}{-1}=\frac{11}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{9}{-1}x=\frac{11}{-1}$$

Whakawehe 9 ki te -1.

$$x^{2}-9x=\frac{11}{-1}$$

Whakawehe 11 ki te -1.

$$x^{2}-9x=-11$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-11+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-9x+\frac{81}{4}=-11+\frac{81}{4}$$

Tāpiri -11 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{37}{4}$$

Tauwehea x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{37}{4}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{\sqrt{37}}{2} x-\frac{9}{2}=-\frac{\sqrt{37}}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{37}+9}{2} x=\frac{9-\sqrt{37}}{2}$$