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

The correct option is A If $a^2-2b<0$, then the equation has one real and two imaginary roots

Let $y=f\left(x\right)=$ $x^3+a{x}^2+bx+c=0$

$dy/dx$ $=f^{\prime }\left(x\right)=$ $3x^2+2ax+b$

Discriminant of f'(x) = $4a^2-12b=4(a^2-3b)$

If $a^2-3b<0$, Then f'(x) has unreal roots and there are no points of local maxima or minima.

Therefore, the graph of f(x) is always increasing and you have only one real root. By Rolle's Theorem.

$a^2-2b<0=>a^2<2b$ is a subset of $a^2-3b<0.$

Therefore, for $a^2-2b<0$, the equation has one real and two imaginary roots.