Question

If $a,b,c\in R$ and $3b^2-8ac<0$, then the equation $a{x}^4+b{x}^3+c{x}^2+5x+7=0$ has

• A. all real roots
• B. all are imaginary roots
• C. can not have all real roots
• D. exactly two real and two imaginary roots

Answer 1

The correct option is C can not have all real roots

Let $f\left(x\right)=a{x}^4+b{x}^3+c{x}^2+5x+7$

Now $f^{\prime }\left(x\right)=4a{x}^3+3b{x}^2+2cx+5$

$f"\left(x\right)=12a{x}^2+6bx+2c$

Now $f"\left(x\right)=0$ implies

$12a{x}^2+6bx+2c=0$

$6a{x}^2+3bx+c=0$

For real roots $D>0$

$9b^2-24ac>0$
$3b^2-8ac>0$.

However, it is given that $3b^2-8ac<0$

Hence, there are no real roots to the equation $f"\left(x\right)=0$

Hence, $f\left(x\right)$ cannot have all roots as real.