If each pair of the three equations `x^(2)+ax+b=0, x^(2)+cx+d=0` and `x^(2)+ex+f=0` has exactly one root in common then show that `(a+c+e)^(2)=4(ac)+ce+ea-b-d-f`

Explanation

To solve the problem, we need to show that if each pair of the three equations \(x^2 + ax + b = 0\), \(x^2 + cx + d = 0\), and \(x^2 + ex + f = 0\) has exactly one root in common, then the following equation holds:

\[ (a + c + e)^2 = 4(ac) + ce + ea - b - d - f \]

### Step-by-Step Solution:

...