Solve -1/2x+k=-1/2left(x+2kright)left(x-5right)

$$- \frac{ 1 }{ 2 } x+k = - \frac{ 1 }{ 2 } \left( x+2k \right) \left( x-5 \right)$$

Solve for k

$$k=-\frac{x\left(x-6\right)}{2\left(x-4\right)}$$

$$x\neq 4$$

Steps for Solving Linear Equation

Use the distributive property to multiply -\frac{1}{2} by x+2k.

$$-\frac{1}{2}x+k=\left(-\frac{1}{2}x-k\right)\left(x-5\right)$$

Use the distributive property to multiply -\frac{1}{2}x-k by x-5.

$$-\frac{1}{2}x+k=-\frac{1}{2}x^{2}+\frac{5}{2}x-kx+5k$$

Add kx to both sides.

$$-\frac{1}{2}x+k+kx=-\frac{1}{2}x^{2}+\frac{5}{2}x+5k$$

Subtract 5k from both sides.

$$-\frac{1}{2}x+k+kx-5k=-\frac{1}{2}x^{2}+\frac{5}{2}x$$

Combine k and -5k to get -4k.

$$-\frac{1}{2}x-4k+kx=-\frac{1}{2}x^{2}+\frac{5}{2}x$$

Add \frac{1}{2}x to both sides.

$$-4k+kx=-\frac{1}{2}x^{2}+\frac{5}{2}x+\frac{1}{2}x$$

Combine \frac{5}{2}x and \frac{1}{2}x to get 3x.

$$-4k+kx=-\frac{1}{2}x^{2}+3x$$

Combine all terms containing k.

$$\left(-4+x\right)k=-\frac{1}{2}x^{2}+3x$$

The equation is in standard form.

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

Divide both sides by -4+x.

$$\frac{\left(x-4\right)k}{x-4}=\frac{x\left(6-x\right)}{2\left(x-4\right)}$$

Dividing by -4+x undoes the multiplication by -4+x.

$$k=\frac{x\left(6-x\right)}{2\left(x-4\right)}$$

Solve for x

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

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