Solve for x x^2-k/2x=2p

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

Combine $x$ and $\frac{k}{2}$ .

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

Subtract $2 p$ from both sides of the equation.

$x^{2} - \frac{x k}{2} - 2 p = 0$

Multiply through by the least common denominator $2$ , then simplify.

Apply the distributive property.

$2 x^{2} + 2 (- \frac{x k}{2}) + 2 (- 2 p) = 0$

Simplify.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x k}{2}$ into the numerator.

$2 x^{2} + 2 (\frac{- x k}{2}) + 2 (- 2 p) = 0$

Cancel the common factor.

$2 x^{2} + 2 (\frac{- x k}{2}) + 2 (- 2 p) = 0$

Rewrite the expression.

$2 x^{2} - x k + 2 (- 2 p) = 0$

Multiply $- 2$ by $2$ .

$2 x^{2} - x k - 4 p = 0$

Move $x$ .

$2 x^{2} - 1 k x - 4 p = 0$

Reorder $2 x^{2}$ and $- 1 k x$ .

$- 1 k x + 2 x^{2} - 4 p = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 2$ , $b = - k$ , and $c = - 4 p$ into the quadratic formula and solve for $x$ .

$\frac{k \pm \sqrt{{(- k)}^{2} - 4 \cdot (2 \cdot (- 4 p))}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

Apply the product rule to $- k$ .

$x = \frac{k \pm \sqrt{{(- 1)}^{2} k^{2} - 4 \cdot 2 \cdot (- 4 p)}}{2 \cdot 2}$

Raise $- 1$ to the power of $2$ .

$x = \frac{k \pm \sqrt{1 k^{2} - 4 \cdot 2 \cdot (- 4 p)}}{2 \cdot 2}$

Multiply $k^{2}$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 4 \cdot 2 \cdot (- 4 p)}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$x = \frac{k \pm \sqrt{k^{2} - 8 \cdot (- 4 p)}}{2 \cdot 2}$

Multiply $- 8$ by $- 4$ .

$x = \frac{k \pm \sqrt{k^{2} + 32 p}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{k \pm \sqrt{k^{2} + 32 p}}{4}$

The final answer is the combination of both solutions.

$x = \frac{k + \sqrt{k^{2} + 32 p}}{4}$

$x = \frac{k - \sqrt{k^{2} + 32 p}}{4}$