Solve (x-4)(x-k)=x^2-5/3kx+m

$$( x - 4 ) ( x - k ) = x ^ { 2 } - \frac { 5 } { 3 } k x + m$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=\frac{3\left(4x+m\right)}{2\left(x+6\right)}\text{, }&x\neq -6\\k\in \mathrm{C}\text{, }&x=-6\text{ and }m=24\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-4 by x-k.

$$x^{2}-xk-4x+4k=x^{2}-\frac{5}{3}kx+m$$

Add \frac{5}{3}kx to both sides.

$$x^{2}-xk-4x+4k+\frac{5}{3}kx=x^{2}+m$$

Combine -xk and \frac{5}{3}kx to get \frac{2}{3}xk.

$$x^{2}+\frac{2}{3}xk-4x+4k=x^{2}+m$$

Subtract x^{2} from both sides.

$$\frac{2}{3}xk-4x+4k=x^{2}+m-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$\frac{2}{3}xk-4x+4k=m$$

Add 4x to both sides.

$$\frac{2}{3}xk+4k=m+4x$$

Combine all terms containing k.

$$\left(\frac{2}{3}x+4\right)k=m+4x$$

The equation is in standard form.

$$\left(\frac{2x}{3}+4\right)k=4x+m$$

Divide both sides by \frac{2}{3}x+4.

$$\frac{\left(\frac{2x}{3}+4\right)k}{\frac{2x}{3}+4}=\frac{4x+m}{\frac{2x}{3}+4}$$

Dividing by \frac{2}{3}x+4 undoes the multiplication by \frac{2}{3}x+4.

$$k=\frac{4x+m}{\frac{2x}{3}+4}$$

Divide m+4x by \frac{2}{3}x+4.

$$k=\frac{3\left(4x+m\right)}{2\left(x+6\right)}$$

Solve for k

$$\left\{\begin{matrix}k=\frac{3\left(4x+m\right)}{2\left(x+6\right)}\text{, }&x\neq -6\\k\in \mathrm{R}\text{, }&x=-6\text{ and }m=24\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-4 by x-k.

$$x^{2}-xk-4x+4k=x^{2}-\frac{5}{3}kx+m$$

Add \frac{5}{3}kx to both sides.

$$x^{2}-xk-4x+4k+\frac{5}{3}kx=x^{2}+m$$

Combine -xk and \frac{5}{3}kx to get \frac{2}{3}xk.

$$x^{2}+\frac{2}{3}xk-4x+4k=x^{2}+m$$

Subtract x^{2} from both sides.

$$\frac{2}{3}xk-4x+4k=x^{2}+m-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$\frac{2}{3}xk-4x+4k=m$$

Add 4x to both sides.

$$\frac{2}{3}xk+4k=m+4x$$

Combine all terms containing k.

$$\left(\frac{2}{3}x+4\right)k=m+4x$$

The equation is in standard form.

$$\left(\frac{2x}{3}+4\right)k=4x+m$$

Divide both sides by \frac{2}{3}x+4.

$$\frac{\left(\frac{2x}{3}+4\right)k}{\frac{2x}{3}+4}=\frac{4x+m}{\frac{2x}{3}+4}$$

Dividing by \frac{2}{3}x+4 undoes the multiplication by \frac{2}{3}x+4.

$$k=\frac{4x+m}{\frac{2x}{3}+4}$$

Divide m+4x by \frac{2}{3}x+4.

$$k=\frac{3\left(4x+m\right)}{2\left(x+6\right)}$$

Solve for m

$$m=\frac{2\left(kx-6x+6k\right)}{3}$$

Solution Steps

Use the distributive property to multiply x-4 by x-k.

$$x^{2}-xk-4x+4k=x^{2}-\frac{5}{3}kx+m$$

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

$$x^{2}-\frac{5}{3}kx+m=x^{2}-xk-4x+4k$$

Subtract x^{2} from both sides.

$$-\frac{5}{3}kx+m=x^{2}-xk-4x+4k-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-\frac{5}{3}kx+m=-xk-4x+4k$$

Add \frac{5}{3}kx to both sides.

$$m=-xk-4x+4k+\frac{5}{3}kx$$

Combine -xk and \frac{5}{3}kx to get \frac{2}{3}xk.

$$m=\frac{2}{3}xk-4x+4k$$