Solve x=kx^2+(1-3x)x+2k-5

$$x = k x ^ { 2 } + ( 1 - 3 x ) x + 2 k - 5$$

Solve for k (complex solution)

$$k=\frac{3x^{2}+5}{x^{2}+2}$$

$$x\neq -\sqrt{2}i\text{ and }x\neq \sqrt{2}i$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1-3x by x.

$$x=kx^{2}+x-3x^{2}+2k-5$$

Swap sides so that all variable terms are on the left hand side.

$$kx^{2}+x-3x^{2}+2k-5=x$$

Subtract x from both sides.

$$kx^{2}-3x^{2}+2k-5=x-x$$

Combine x and -x to get 0.

$$kx^{2}-3x^{2}+2k-5=0$$

Add 3x^{2} to both sides. Anything plus zero gives itself.

$$kx^{2}+2k-5=3x^{2}$$

Add 5 to both sides.

$$kx^{2}+2k=3x^{2}+5$$

Combine all terms containing k.

$$\left(x^{2}+2\right)k=3x^{2}+5$$

Divide both sides by x^{2}+2.

$$\frac{\left(x^{2}+2\right)k}{x^{2}+2}=\frac{3x^{2}+5}{x^{2}+2}$$

Dividing by x^{2}+2 undoes the multiplication by x^{2}+2.

$$k=\frac{3x^{2}+5}{x^{2}+2}$$

Solve for k

$$k=\frac{3x^{2}+5}{x^{2}+2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1-3x by x.

$$x=kx^{2}+x-3x^{2}+2k-5$$

Swap sides so that all variable terms are on the left hand side.

$$kx^{2}+x-3x^{2}+2k-5=x$$

Subtract x from both sides.

$$kx^{2}-3x^{2}+2k-5=x-x$$

Combine x and -x to get 0.

$$kx^{2}-3x^{2}+2k-5=0$$

Add 3x^{2} to both sides. Anything plus zero gives itself.

$$kx^{2}+2k-5=3x^{2}$$

Add 5 to both sides.

$$kx^{2}+2k=3x^{2}+5$$

Combine all terms containing k.

$$\left(x^{2}+2\right)k=3x^{2}+5$$

Divide both sides by x^{2}+2.

$$\frac{\left(x^{2}+2\right)k}{x^{2}+2}=\frac{3x^{2}+5}{x^{2}+2}$$

Dividing by x^{2}+2 undoes the multiplication by x^{2}+2.

$$k=\frac{3x^{2}+5}{x^{2}+2}$$

Solve for x (complex solution)

$$x=-i\left(3-k\right)^{-\frac{1}{2}}\sqrt{5-2k}$$

$$x=i\left(3-k\right)^{-\frac{1}{2}}\sqrt{5-2k}\text{, }k\neq 3$$

Solve for x

$$x=\sqrt{\frac{2k-5}{3-k}}$$

$$x=-\sqrt{\frac{2k-5}{3-k}}\text{, }k\geq \frac{5}{2}\text{ and }k<3$$