Discuss the nature of roots of equation ` x^(2) +ax +b =0 and x^(2) +cx + d=0, where a,b,c, d are real numbers and `ac = 2( b +d)`

Explanation

Let ` D_(1) and D_(2)` be discriminants of the given equations ` x^(2) +ax + b =0 and x^(2) +cx +d =0` respectively. ltbr. Then ` D_(1)=a^(2) -4b and D_(2) =c^(2)-4d` .
Now ` D_(1) + D_(2) =a^(2) -4b + c^(2) -4d`
` a^(2) +c^(2) -2ac [ therefore , ac = 2 ( b +d)]`
` = (a -c)^(2) ge 0`
Consequently at least one of the ficen equations has real roots.