Question

Consider the cubic equation $x^3+a{x}^2+bx+c=0$, where $a,b,c$ are real numbers. Which of the following statements is correct?

• A. If $a^2-2b<0$, then the equation has one real and two imaginary roots
• B. If $a^2-2b\ge 0$, then the equation has all real roots
• C. If $a^2-2b>0$, then the equation has all real and distinct roots
• D. If $4a^3-27b^2>0$, then the equation has real and distinct roots

Answer 1

Answer: (A)

We have, $x^3+a{x}^2+bx+c=0$
Let $f(x)=x^3+a{x}^2+bx+c$
$P^{\prime }(x)=3x^2+2ax+b$
$D=(2a{)}^2-4(3)(b)$
$D=4a^2-12b$
$D=4(a^2-3b)$
For three roots $a^2-3b>0$
but given $a^2-2b<0$
$\therefore D<0$
Hence, $f(x)$ has one real roots and two imaginary roots