Whakaoti i te left(1-kright)x^2+x+1-k=0

$$\left( 1-k \right) { x }^{ 2 } +x+1-k = 0$$

Whakaoti mō k (complex solution)

$$k=\frac{x^{2}+x+1}{x^{2}+1}$$

$$x\neq -i\text{ and }x\neq i$$

Ngā Upane mō te Whakaoti Whārite Rārangi

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

$$x^{2}-kx^{2}+x+1-k=0$$

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

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

Tangohia te x mai i ngā taha e rua.

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

Tangohia te 1 mai i ngā taha e rua.

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

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

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

Whakawehea ngā taha e rua ki te -x^{2}-1.

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

Mā te whakawehe ki te -x^{2}-1 ka wetekia te whakareanga ki te -x^{2}-1.

$$k=\frac{-x^{2}-x-1}{-x^{2}-1}$$

Whakawehe -x^{2}-x-1 ki te -x^{2}-1.

$$k=\frac{x^{2}+x+1}{x^{2}+1}$$

Whakaoti mō k

$$k=\frac{x^{2}+x+1}{x^{2}+1}$$

Ngā Upane mō te Whakaoti Whārite Rārangi

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

$$x^{2}-kx^{2}+x+1-k=0$$

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

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

Tangohia te x mai i ngā taha e rua.

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

Tangohia te 1 mai i ngā taha e rua.

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

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

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

Whakawehea ngā taha e rua ki te -x^{2}-1.

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

Mā te whakawehe ki te -x^{2}-1 ka wetekia te whakareanga ki te -x^{2}-1.

$$k=\frac{-x^{2}-x-1}{-x^{2}-1}$$

Whakawehe -x^{2}-x-1 ki te -x^{2}-1.

$$k=\frac{x^{2}+x+1}{x^{2}+1}$$

Whakaoti mō x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{\left(3-2k\right)\left(-2\left(1-k\right)+1\right)}-1}{2\left(1-k\right)}\text{; }x=\frac{\sqrt{-4k^{2}+8k-3}+1}{2\left(k-1\right)}\text{, }&k\neq 1\\x=0\text{, }&k=1\end{matrix}\right.$$

Whakaoti mō x

$$\left\{\begin{matrix}x=\frac{\sqrt{-\left(1-2k\right)\left(3-2k\right)}-1}{2\left(1-k\right)}\text{; }x=-\frac{\sqrt{-\left(1-2k\right)\left(3-2k\right)}+1}{2\left(1-k\right)}\text{, }&k\neq 1\text{ and }k\leq \frac{3}{2}\text{ and }k\geq \frac{1}{2}\\x=0\text{, }&k=1\end{matrix}\right.$$