Question

The quadratic equation abx2 + acx + b(bx + c) = 0

has non-zero equal and rational roots. The values of

a and c respectively cannot be equal to (ab  0)

Answer 1

quadratic equation is abx² + acx + b(bx + c) = 0
or, abx² + acx + b²x + bc = 0

A/C to question,
roots are equal and rational numbers .
so, discriminant = 0
(ac + b²)² - 4(ab)(bc) = 0
a²c² + b⁴ + 2ab²c - 4ab²c = 0
a²c² + b⁴ - 2ab²c = 0
(ac)² + (b²)² - 2(ac)(b²) = 0
(ac - b²)² = 0
so, ac = b²

we get the condition is ac = b²
here you should remember that roots will be rational number only when coefficient of roots also will be rational number.
so, a , b and c are also rational numbers.

check option (d)
8 and 49
ac = 8 × 49 = b² => b = ±14√2 {irradiation number}
but b is also a rational number so, option (d) is not satisfied the above condition.
hence, a and c respectively cannot equal to 8 and 49.

so, option (d) is correct.