Question

In the following, determine whether the given quadratic equation have real roots and if so, find the roots :
$\sqrt{3}{x}^2+10x-8\sqrt{3}=0$

• A. $-4\sqrt{3},\frac{2}{\sqrt{3}}$
• B. $4\sqrt{3},\frac{2}{\sqrt{3}}$
• C. $-4\sqrt{3},\frac{-2}{\sqrt{3}}$
• D. not real

Answer 1

Answer: (A)

$-4\sqrt{3},\frac{2}{\sqrt{3}}$

$\sqrt{3}{x}^2+10x-8\sqrt{3}=0$

$\Rightarrow$ $\sqrt{3}{x}^2+12x-2x-8\sqrt{3}=0$

$\Rightarrow$ $\sqrt{3}x(x+4\sqrt{3})-2(x+4\sqrt{3})=0$

$\Rightarrow$ $(x+4\sqrt{3})(\sqrt{3}x-2)=0$

$\Rightarrow$ $x+4\sqrt{3}=0$ and $\sqrt{3}x-2=0$

$\Rightarrow$ $x=-4\sqrt{3}$ and $x=\frac{2}{\sqrt{3}}$

$\therefore$ The given quadratic equation has real roots which are $-4\sqrt{3},\frac{2}{\sqrt{3}}$.