Whakaoti i te (3k+1)x^2+3k-1/(3k+1)(3k^2+8k-3)+x/9k^2-1=1/3k^2+10k+3

$$\frac { ( 3 k + 1 ) x ^ { 2 } + 3 k - 1 } { ( 3 k + 1 ) ( 3 k ^ { 2 } + 8 k - 3 ) } + \frac { x } { 9 k ^ { 2 } - 1 } = \frac { 1 } { 3 k ^ { 2 } + 10 k + 3 }$$

Whakaoti mō k (complex solution)

$$\left\{\begin{matrix}k=-\frac{x+3}{3x+1}\text{, }&x\neq -\frac{1}{3}\text{ and }x\neq 0\text{ and }x\neq -\frac{5}{3}\\k\in \mathrm{C}\setminus -\frac{1}{3},-3,\frac{1}{3}\text{, }&x=0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe k ki tētahi o ngā uara -3,-\frac{1}{3},\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(3k-1\right)\left(k+3\right)\left(3k+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o \left(3k+1\right)\left(3k^{2}+8k-3\right),9k^{2}-1,3k^{2}+10k+3.

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

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te k+3 ki te x.

$$3kx^{2}+x^{2}+3k-1+kx+3x=3k-1$$

Tangohia te 3k mai i ngā taha e rua.

$$3kx^{2}+x^{2}+3k-1+kx+3x-3k=-1$$

Pahekotia te 3k me -3k, ka 0.

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

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

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

Me tāpiri te 1 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 1, ka 0.

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

Tangohia te 3x mai i ngā taha e rua.

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

Pahekotia ngā kīanga tau katoa e whai ana i te k.

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

Whakawehea ngā taha e rua ki te 3x^{2}+x.

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

Mā te whakawehe ki te 3x^{2}+x ka wetekia te whakareanga ki te 3x^{2}+x.

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

Whakawehe -x\left(3+x\right) ki te 3x^{2}+x.

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

Tē taea kia ōrite te tāupe k ki tētahi o ngā uara -\frac{1}{3},-3,\frac{1}{3}.

$$k=-\frac{x+3}{3x+1}\text{, }k\neq -\frac{1}{3}\text{ and }k\neq -3\text{ and }k\neq \frac{1}{3}$$

Whakaoti mō k

$$\left\{\begin{matrix}k=-\frac{x+3}{3x+1}\text{, }&x\neq -\frac{1}{3}\text{ and }x\neq -\frac{5}{3}\text{ and }x\neq 0\\k\in \mathrm{R}\setminus -\frac{1}{3},\frac{1}{3},-3\text{, }&x=0\end{matrix}\right.$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe k ki tētahi o ngā uara -3,-\frac{1}{3},\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(3k-1\right)\left(k+3\right)\left(3k+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o \left(3k+1\right)\left(3k^{2}+8k-3\right),9k^{2}-1,3k^{2}+10k+3.

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

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te k+3 ki te x.

$$3kx^{2}+x^{2}+3k-1+kx+3x=3k-1$$

Tangohia te 3k mai i ngā taha e rua.

$$3kx^{2}+x^{2}+3k-1+kx+3x-3k=-1$$

Pahekotia te 3k me -3k, ka 0.

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

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

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

Me tāpiri te 1 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 1, ka 0.

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

Tangohia te 3x mai i ngā taha e rua.

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

Pahekotia ngā kīanga tau katoa e whai ana i te k.

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

Whakawehea ngā taha e rua ki te 3x^{2}+x.

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

Mā te whakawehe ki te 3x^{2}+x ka wetekia te whakareanga ki te 3x^{2}+x.

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

Whakawehe -x\left(3+x\right) ki te 3x^{2}+x.

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

Tē taea kia ōrite te tāupe k ki tētahi o ngā uara -\frac{1}{3},-3,\frac{1}{3}.

$$k=-\frac{x+3}{3x+1}\text{, }k\neq -\frac{1}{3}\text{ and }k\neq -3\text{ and }k\neq \frac{1}{3}$$

Whakaoti mō x (complex solution)

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

$$x=0\text{, }k\neq -\frac{1}{3}\text{ and }k\neq -3\text{ and }k\neq \frac{1}{3}$$

Whakaoti mō x

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

$$x=0\text{, }k\neq -3\text{ and }|k|\neq \frac{1}{3}$$