Question

If the roots of the quadratic equation `ax^2+bx+c=0` are imaginary then for all values of `a,b,c` and `x in R` then the nature of expression `a^2x^2+abx+ac` is?

A. Positive
B. non-negative
C Negative
D may be positive, zero or negative
Ans is A

Answer 1

See explanation...

Explanation:

To say that the roots of `ax^2+bx+c=0` are imaginary is to say that the graph of:

`f(x) = ax^2+bx+c`

does not intersect the `x` axis for any real value of `x`.

Since this is described as a quadratic, we must have `a!=0`.

So the whole of the parabola either lies on one side of the `x` axis or the other.

When we multiply by `a!=0`, getting `a^2x^2+abx+ac` whose graph is simply a scaled version of the original, the coefficient of `x^2` is `a^2 > 0`. Hence we know that for large values of `x` the quadratic function will be positive. Hence it is positive for all values of `x`.

So the answer is indeed A.