find the value of x : 1 /(p+q+x) = 1/p +1/q + 1/ x
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This gives : 1/p + 1/q = 1/(p+q+x) - 1/x
=> (q+p) / (pq) = {x - (p+q+x)} / {(p+q+x).x}
=> (p+q) / (pq) = - (p+q) / {(p+q+x).x}
=> 1/(pq) = -1 / {(p+q+x).x} ( since, p+q ≠ 0 )
=> (p+q+x).x = - pq
=> px +qx + x^2 + pq = 0
=> x^2 + px + qx + pq = 0
=> x ( x + p ) + q ( x + p ) = 0
=> ( x + p ) . ( x + q ) = 0
=> x + p = 0 OR x + q = 0
=> x = - p OR x = - q
=> (q+p) / (pq) = {x - (p+q+x)} / {(p+q+x).x}
=> (p+q) / (pq) = - (p+q) / {(p+q+x).x}
=> 1/(pq) = -1 / {(p+q+x).x} ( since, p+q ≠ 0 )
=> (p+q+x).x = - pq
=> px +qx + x^2 + pq = 0
=> x^2 + px + qx + pq = 0
=> x ( x + p ) + q ( x + p ) = 0
=> ( x + p ) . ( x + q ) = 0
=> x + p = 0 OR x + q = 0
=> x = - p OR x = - q
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