Question

1/p+ 1 / q + 1 / x is equal to 1 / x+ p + q take :(1 / p+ 1 / q)+ (1 / X - 1 / X + P + Q)=0 solve the quadratic ​

Answer 1

Given equation,

$\sf{ \frac{1}{p} + \frac{1}{q} + \frac{1}{x} = \frac{1}{p + q + x} }$

To find the roots of the equation,

On transposing,

$\sf{ \frac{1}{p} + \frac{1}{q} = \frac{1}{p + q + x} - \frac{1}{x} } \\ \\ \implies \: \sf { \frac{p + q}{pq} = \frac{x - (p + q + x)}{x(p + q + x)} } \\ \\ \implies \: \sf{ \frac{p + q}{pq} = \frac{ - (p + q)}{x {}^{2} + px + qx} } \\ \\ \implies \: \sf{ \frac{1}{pq} = \frac{ - 1}{x {}^{2} + px + qx } } \\ \\ \implies \: \sf{x {}^{2} + px + qx = - pq } \\ \\ \implies \: \sf{x {}^{2} + px + qx + pq = 0 } \\ \\ \implies \: \sf{x(x + p) + q(x + p) = 0} \\ \\ \implies \: \sf{(x + p)(x + q) = 0} \\ \\ \implies \: \: \huge{\boxed{{\sf{x = - p \: \: or \: \: - q }}}}$

The zeros of the equation are -p and -q