Question

If a, b $\epsilon$ R, a $\ne$ 0 and the quadratic equation $a{x}^2-bx+2$ =0 has imaginary roots, then a + b + 2 is

• A. Negative
• B. Positive
• C. Zero
• D. Depends on sign of b

Answer 1

The correct option is B

Positive

Observe that for given expression $f\left(x\right)$ = $a{x}^2-bx+2$

$f(-1)$ = $a+b+2$

$f\left(x\right)$ = 0 has imaginary roots

So, $b^2-4ac$ < 0

$b^2$ < 4ac

$b^2$ is always > 0 and for 4ac to be greater than $b^2$

4ac > 0 $\Rightarrow$ac > 0

$\Rightarrow$ a & c should be of same sign

Given c = 2(> 0) hence a > 0

As a > 0 & D < 0, graph looks as below

Hence, for $\forall$ x $\epsilon$ R, f(x) is positive

Hence f(-1) is also positive