Question

The nature of the roots of the equation $8x^3+36x^2+54x+27=0$ is:

• A. One real and two imaginary
• B. All real and distinct
• C. Real and equal
• D. Real and positive

Answer 1

The correct option is C Real and equal

Let $f\left(x\right)=8x^3+36x^2+54x+27$
$\Rightarrow f^{\prime }\left(x\right)=24x^2+72x+54$
$\Rightarrow f^{\prime }\left(x\right)=0\Rightarrow x=-\frac{3}{2},-\frac{3}{2}$
$f^{\prime \prime }\left(x\right)=48x+72$
Clearly, $f^{\prime \prime }(-\frac{3}{2})=0=f(-\frac{3}{2})$
$\therefore$ Roots of $f\left(x\right)=0$ are real and equal.