Question

Solve the equation $x^2-\sqrt{3}x+1=0$:

• A. $x=\frac{(-\sqrt{3}\pm 2i)}{2}$
• B. $x=\frac{\left(-\sqrt{3}\pm i\right)}{2}$
• C. $x=\left(\sqrt{3}\pm i\right)$
• D. $x=\frac{\left(\sqrt{3}\pm i\right)}{2}$

Answer 1

The correct option is D

$x=\frac{\left(\sqrt{3}\pm i\right)}{2}$

Explanation for correct option

Finding the roots of a quadratic equation:

Given quadratic equation is $x^2-\sqrt{3}x+1=0$.

Compare the above equation with the standard form: $a{x}^2+bx+c=0$.

We get, $a=1,b=-\sqrt{3},c=1$

The roots of the standard form of a quadratic equation is given by:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Hence, the required roots are:

$$\begin{gathered} x=\frac{-\left(-\sqrt{3}\right)\pm \sqrt{{\left(-\sqrt{3}\right)}^2-4\left(1\right)\left(1\right)}}{2\left(1\right)} \\ =\frac{\sqrt{3}\pm \sqrt{3-4}}{2} \\ =\frac{\sqrt{3}\pm \sqrt{-1}}{2}\left[\because \sqrt{-1}=i\right] \\ =\frac{\sqrt{3}\pm i}{2} \end{gathered}$$

Therefore, the correct answer is Option (D).