Question

Consider the cubic equation $x^{3} + ax^{2} + bx+ c = 0$, 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, $x^{3}+ax^{2}+bx+c=0$
Let $f (x) =x^{3}+ax^{2}+bx+c$
$P'(x) =3x^{2}+2ax+b$
$D=(2a)^{2}-4 (3) \,(b)$
$D=4a^{2}-12b$
$D=4(a^{2}-3b)$
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