How to solve by quadratic formula: (p-q)x+(2q)/x=(p+q) ?

1 Answer
Nov 21, 2017

x=1" " or " "x = (2q)/(p-q)

Explanation:

Given:

(p-q)x + (2q)/x = (p+q)

Multiply both sides of the equation by x to get:

(p-q)x^2+2q = (p+q)x

Subtract (p+q)x from both sides to get:

(p-q)x^2-(p+q)x+2q = 0

This is in standard form:

ax^2+bx+c = 0

with a=(p-q), b=-(p+q) and c=2q

It has roots given by the quadratic formula:

x = (-b+-sqrt(b^2-4ac))/(2a)

color(white)(x) = ((p+q)+-sqrt((-(p+q))^2-4(p-q)(2q)))/(2(p-q))

color(white)(x) = ((p+q)+-sqrt((p^2+2pq+q^2)-(8pq-8q^2)))/(2(p-q))

color(white)(x) = ((p+q)+-sqrt(p^2-6pq+9q^2))/(2(p-q))

color(white)(x) = ((p+q)+-sqrt((p-3q)^2))/(2(p-q))

color(white)(x) = ((p+q)+-(p-3q))/(2(p-q))

That is:

x = ((p+q)+(p-3q))/(2(p-q)) = (2p-2q)/(2p-2q) = 1

or:

x = ((p+q)-(p-3q))/(2(p-q)) = (4q)/(2(p-q)) = (2q)/(p-q)