Solve for x (2x+p)/q=(2x+q)/p

$\frac{2 x + p}{q} = \frac{2 x + q}{p}$

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(2 x + p) p = q (2 x + q)$

Solve the equation for $x$ .

Simplify $(2 x + p) p$ .

Start simplifying.

$0 + 0 + (2 x + p) p = q (2 x + q)$

Simplify by adding zeros.

$(2 x + p) p = q (2 x + q)$

Apply the distributive property.

$2 x p + p \cdot p = q (2 x + q)$

Multiply $p$ by $p$ .

$2 x p + p^{2} = q (2 x + q)$

Simplify $q (2 x + q)$ .

Apply the distributive property.

$2 x p + p^{2} = q (2 x) + q \cdot q$

Simplify the expression.

Rewrite using the commutative property of multiplication.

$2 x p + p^{2} = 2 q x + q \cdot q$

Multiply $q$ by $q$ .

$2 x p + p^{2} = 2 q x + q^{2}$

Subtract $2 q x$ from both sides of the equation.

$2 x p + p^{2} - 2 q x = q^{2}$

Subtract $p^{2}$ from both sides of the equation.

$2 x p - 2 q x = q^{2} - p^{2}$

Factor $2 x$ out of $2 x p - 2 q x$ .

Factor $2 x$ out of $2 x p$ .

$2 x (p) - 2 q x = q^{2} - p^{2}$

Factor $2 x$ out of $- 2 q x$ .

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

Factor $2 x$ out of $2 x (p) + 2 x (- q)$ .

$2 x (p - q) = q^{2} - p^{2}$

Divide each term in $2 x (p - q) = q^{2} - p^{2}$ by $2 (p - q)$ and simplify.

Divide each term in $2 x (p - q) = q^{2} - p^{2}$ by $2 (p - q)$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $p - q$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{q^{2}}{2 (p - q)} - \frac{p^{2}}{2 (p - q)}$

Solve for x (2x+p)/q=(2x+q)/p

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