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

1 Answer
George C.
Jul 18, 2017

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.

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