Question

The roots of equation $x^3-6x+9=0$ is

• A. $-6$
• B. $-9$
• C. $6$
• D. $-3$

Answer 1

The correct option is D

$-3$

Explanation for the correct option:

Find the roots of the given equation

Given equation,

$$\begin{gathered} x^3-6x+9=0 \\ \Rightarrow x^3+3x^2-3x^2-9x+3x+9=0 \\ \Rightarrow x^2\left(x+3\right)-3x\left(x+3\right)+3\left(x+3\right)=0 \\ \Rightarrow \left(x+3\right)\left(x^2-3x+3\right)=0 \end{gathered}$$

Now root of the quadratic equation $x^2-3x+3$is

$$\begin{gathered} x=\frac{3\pm \sqrt{9-12}}{2} \\ =\frac{3\pm i\sqrt{3}}{2}\left[\because i=\sqrt{-1}\right] \end{gathered}$$

Therefore, there is only one real root and that is $-3$

Hence, the correct option is $\left(\mathrm{D}\right)$.