Whakaoti i te 3/x+(6x+5)-6x+1/3x-1=3x+9x+6/3x-1+3x^2+1/x

$$\frac { 3 } { x } + ( 6 x + 5 ) - \frac { 6 x + 1 } { 3 x - 1 } = 3 x + \frac { 9 x + 6 } { 3 x - 1 } + \frac { 3 x ^ { 2 } + 1 } { x }$$

Whakaoti mō x

$$x=-\frac{1}{3}\approx -0.333333333$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(3x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,3x-1.

$$\left(3x-1\right)\times 3+6xx\left(3x-1\right)+x\left(3x-1\right)\times 5-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x-1 ki te 3.

$$9x-3+6xx\left(3x-1\right)+x\left(3x-1\right)\times 5-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$9x-3+6x^{2}\left(3x-1\right)+x\left(3x-1\right)\times 5-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6x^{2} ki te 3x-1.

$$9x-3+18x^{3}-6x^{2}+x\left(3x-1\right)\times 5-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 3x-1.

$$9x-3+18x^{3}-6x^{2}+\left(3x^{2}-x\right)\times 5-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2}-x ki te 5.

$$9x-3+18x^{3}-6x^{2}+15x^{2}-5x-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Pahekotia te -6x^{2} me 15x^{2}, ka 9x^{2}.

$$9x-3+18x^{3}+9x^{2}-5x-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

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

$$4x-3+18x^{3}+9x^{2}-x\left(6x+1\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 6x+1.

$$4x-3+18x^{3}+9x^{2}-\left(6x^{2}+x\right)=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Hei kimi i te tauaro o 6x^{2}+x, kimihia te tauaro o ia taurangi.

$$4x-3+18x^{3}+9x^{2}-6x^{2}-x=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Pahekotia te 9x^{2} me -6x^{2}, ka 3x^{2}.

$$4x-3+18x^{3}+3x^{2}-x=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Pahekotia te 4x me -x, ka 3x.

$$3x-3+18x^{3}+3x^{2}=3xx\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$3x-3+18x^{3}+3x^{2}=3x^{2}\left(3x-1\right)+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2} ki te 3x-1.

$$3x-3+18x^{3}+3x^{2}=9x^{3}-3x^{2}+x\left(9x+6\right)+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 9x+6.

$$3x-3+18x^{3}+3x^{2}=9x^{3}-3x^{2}+9x^{2}+6x+\left(3x-1\right)\left(3x^{2}+1\right)$$

Pahekotia te -3x^{2} me 9x^{2}, ka 6x^{2}.

$$3x-3+18x^{3}+3x^{2}=9x^{3}+6x^{2}+6x+\left(3x-1\right)\left(3x^{2}+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x-1 ki te 3x^{2}+1.

$$3x-3+18x^{3}+3x^{2}=9x^{3}+6x^{2}+6x+9x^{3}+3x-3x^{2}-1$$

Pahekotia te 9x^{3} me 9x^{3}, ka 18x^{3}.

$$3x-3+18x^{3}+3x^{2}=18x^{3}+6x^{2}+6x+3x-3x^{2}-1$$

Pahekotia te 6x me 3x, ka 9x.

$$3x-3+18x^{3}+3x^{2}=18x^{3}+6x^{2}+9x-3x^{2}-1$$

Pahekotia te 6x^{2} me -3x^{2}, ka 3x^{2}.

$$3x-3+18x^{3}+3x^{2}=18x^{3}+3x^{2}+9x-1$$

Tangohia te 18x^{3} mai i ngā taha e rua.

$$3x-3+18x^{3}+3x^{2}-18x^{3}=3x^{2}+9x-1$$

Pahekotia te 18x^{3} me -18x^{3}, ka 0.

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

Tangohia te 3x^{2} mai i ngā taha e rua.

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

Pahekotia te 3x^{2} me -3x^{2}, ka 0.

$$3x-3=9x-1$$

Tangohia te 9x mai i ngā taha e rua.

$$3x-3-9x=-1$$

Pahekotia te 3x me -9x, ka -6x.

$$-6x-3=-1$$

Me tāpiri te 3 ki ngā taha e rua.

$$-6x=-1+3$$

Tāpirihia te -1 ki te 3, ka 2.

$$-6x=2$$

Whakawehea ngā taha e rua ki te -6.

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

Whakahekea te hautanga \frac{2}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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