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

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

Whakaoti mō x

$$x=\sqrt{5}+2\approx 4.236067977$$

$$x=2-\sqrt{5}\approx -0.236067977$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2\left(x-1\right).

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

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

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

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

$$6x+2=2x^{2}-x\times 2$$

Whakareatia te -1 ki te 2, ka -2.

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

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

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

Me tāpiri te 2x ki ngā taha e rua.

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

Pahekotia te 6x me 2x, ka 8x.

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

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.

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

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

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

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\left(-2\right)\times 2}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-8±\sqrt{64+8\times 2}}{2\left(-2\right)}$$

Whakareatia 8 ki te 2.

$$x=\frac{-8±\sqrt{64+16}}{2\left(-2\right)}$$

Tāpiri 64 ki te 16.

$$x=\frac{-8±\sqrt{80}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 80.

$$x=\frac{-8±4\sqrt{5}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-8±4\sqrt{5}}{-4}$$

Nā, me whakaoti te whārite x=\frac{-8±4\sqrt{5}}{-4} ina he tāpiri te ±. Tāpiri -8 ki te 4\sqrt{5}.

$$x=\frac{4\sqrt{5}-8}{-4}$$

Whakawehe -8+4\sqrt{5} ki te -4.

$$x=2-\sqrt{5}$$

Nā, me whakaoti te whārite x=\frac{-8±4\sqrt{5}}{-4} ina he tango te ±. Tango 4\sqrt{5} mai i -8.

$$x=\frac{-4\sqrt{5}-8}{-4}$$

Whakawehe -8-4\sqrt{5} ki te -4.

$$x=\sqrt{5}+2$$

Kua oti te whārite te whakatau.

$$x=2-\sqrt{5} x=\sqrt{5}+2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2\left(x-1\right).

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

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

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

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

$$6x+2=2x^{2}-x\times 2$$

Whakareatia te -1 ki te 2, ka -2.

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

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

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

Me tāpiri te 2x ki ngā taha e rua.

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

Pahekotia te 6x me 2x, ka 8x.

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

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

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.

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

Whakawehea ngā taha e rua ki te -2.

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

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

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

Whakawehe 8 ki te -2.

$$x^{2}-4x=-\frac{2}{-2}$$

Whakawehe -2 ki te -2.

$$x^{2}-4x=1$$

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.

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

Pūrua -2.

$$x^{2}-4x+4=1+4$$

Tāpiri 1 ki te 4.

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

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}.

$$\left(x-2\right)^{2}=5$$

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

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

Whakarūnātia.

$$x-2=\sqrt{5} x-2=-\sqrt{5}$$

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

$$x=\sqrt{5}+2 x=2-\sqrt{5}$$