Question

If the roots of the quadratic equation $a{x}^2+bx+c=0$ are non real, then for all values of $a,b,c$ and $x\in R$, the expression $a^2{x}^2+abx+ac$ is

• A. positive
• B. non-negative
• C. negative
• D. may be positive, zero and negative

Answer 1

The correct option is A positive

Roots of the quadratic equation $a{x}^2+bx+c=0$ are non real.
$\Rightarrow b^2-4ac<0$ -----(1)
Now, for the expression $a^2{x}^2+abx+ac$,
$\Delta =a^2{b}^2-4a^{3}c=a^2(b^2-4ac)<0$ from (1)
Since, discriminant value is less than zero, coefficient of $x^2$ and expression always have same sign.
$\therefore a^2{x}^2+abx+ac>0$
Hence, option A.