Solve x^2-3kx+k+6=00

$$x ^ { 2 } - 3 k x + k + 6 = 00$$

Solve for k

$$k=-\frac{x^{2}+6}{1-3x}$$

$$x\neq \frac{1}{3}$$

Steps for Solving Linear Equation

Multiply 0 and 0 to get 0.

$$x^{2}-3kx+k+6=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-3kx+k+6=-x^{2}$$

Subtract 6 from both sides.

$$-3kx+k=-x^{2}-6$$

Combine all terms containing k.

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

The equation is in standard form.

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

Divide both sides by -3x+1.

$$\frac{\left(1-3x\right)k}{1-3x}=\frac{-x^{2}-6}{1-3x}$$

Dividing by -3x+1 undoes the multiplication by -3x+1.

$$k=\frac{-x^{2}-6}{1-3x}$$

Divide -x^{2}-6 by -3x+1.

$$k=-\frac{x^{2}+6}{1-3x}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{9k^{2}-4k-24}+3k}{2}$$

$$x=\frac{-\sqrt{9k^{2}-4k-24}+3k}{2}$$

Solve for x

$$x=\frac{\sqrt{9k^{2}-4k-24}+3k}{2}$$

$$x=\frac{-\sqrt{9k^{2}-4k-24}+3k}{2}\text{, }k\geq \frac{2\sqrt{55}+2}{9}\text{ or }k\leq \frac{2-2\sqrt{55}}{9}$$