# 1/p+1/q+1/x=1/x+p+q.......there's a clue given but still can't sort it out  (1/p+1/q)+(1/x-1/p+q+x)

sciencesolve | Teacher | (Level 3) Educator Emeritus

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The request of the problem is vague, hence, supposing that you need to solve for x the given equation, you should move `1/x`  to the right such that:

`1/p + 1/q = 1/(x + p + q) - 1/x`

You need to bring the fractions, both sides, to a common denominator such that:

`(q+p)/(pq) = (x - x - p - q)/(x(x+p+q))`

`(q + p)/(pq) = -(p+ q)/(x(x+p+q))`

You may divide by `p+q`  since `p!=0, q!=0=> p+q!=0` , such that:

`1/(pq) = -1/(x(x+p+q)) => -pq = x^2 + xp + xq`

`x^2 + xp + xq + pq = 0`

You need to group the terms such that:

`(x^2 + xp) + (xq + pq) = 0`

Factoring out x in the first group and q in the second group, yields:

`x(x+p) + q(x+p) = 0`

Factoring out `x + p`  yields:

`(x+p)(x+q) = 0 => {(x+p = 0),(x + q = 0):} => {(x=-p),(x=-q):}`

Hence, evaluating the solutions to the given equation yields `x = -p`  and `x = -q` .