Whakaoti i te x+1/x-3=9

$$x + \frac { 1 } { x - 3 } = 9$$

Whakaoti mō x

$$x=2\sqrt{2}+6\approx 8.828427125$$

$$x=6-2\sqrt{2}\approx 3.171572875$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 3 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-3.

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

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

$$x^{2}-3x+1=9\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9 ki te x-3.

$$x^{2}-3x+1=9x-27$$

Tangohia te 9x mai i ngā taha e rua.

$$x^{2}-3x+1-9x=-27$$

Pahekotia te -3x me -9x, ka -12x.

$$x^{2}-12x+1=-27$$

Me tāpiri te 27 ki ngā taha e rua.

$$x^{2}-12x+1+27=0$$

Tāpirihia te 1 ki te 27, ka 28.

$$x^{2}-12x+28=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -12 mō b, me 28 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 28}}{2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 28}}{2}$$

Whakareatia -4 ki te 28.

$$x=\frac{-\left(-12\right)±\sqrt{144-112}}{2}$$

Tāpiri 144 ki te -112.

$$x=\frac{-\left(-12\right)±\sqrt{32}}{2}$$

Tuhia te pūtakerua o te 32.

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

Ko te tauaro o -12 ko 12.

$$x=\frac{12±4\sqrt{2}}{2}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{2}}{2} ina he tāpiri te ±. Tāpiri 12 ki te 4\sqrt{2}.

$$x=\frac{4\sqrt{2}+12}{2}$$

Whakawehe 12+4\sqrt{2} ki te 2.

$$x=2\sqrt{2}+6$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{2}}{2} ina he tango te ±. Tango 4\sqrt{2} mai i 12.

$$x=\frac{12-4\sqrt{2}}{2}$$

Whakawehe 12-4\sqrt{2} ki te 2.

$$x=6-2\sqrt{2}$$

Kua oti te whārite te whakatau.

$$x=2\sqrt{2}+6 x=6-2\sqrt{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 3 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-3.

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

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

$$x^{2}-3x+1=9\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9 ki te x-3.

$$x^{2}-3x+1=9x-27$$

Tangohia te 9x mai i ngā taha e rua.

$$x^{2}-3x+1-9x=-27$$

Pahekotia te -3x me -9x, ka -12x.

$$x^{2}-12x+1=-27$$

Tangohia te 1 mai i ngā taha e rua.

$$x^{2}-12x=-27-1$$

Tangohia te 1 i te -27, ka -28.

$$x^{2}-12x=-28$$

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=-28+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=-28+36$$

Tāpiri -28 ki te 36.

$$x^{2}-12x+36=8$$

Tauwehea x^{2}-12x+36. 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}.

$$\left(x-6\right)^{2}=8$$

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

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

Whakarūnātia.

$$x-6=2\sqrt{2} x-6=-2\sqrt{2}$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=2\sqrt{2}+6 x=6-2\sqrt{2}$$