Question

Factorize $\sqrt{3}{x}^2+5x+2\sqrt{3}=0$ by splitting the middle term and find the value of $x.$

• A. $x=\frac{-2}{\sqrt{3}},x=-\sqrt{3}$
• B. $x=\frac{2}{\sqrt{3}},x=\sqrt{3}$
• C. $x=\frac{\sqrt{2}}{\sqrt{3}},x=-\sqrt{2}$
• D. $x=\frac{-3}{\sqrt{2}},x=\sqrt{3}$

Answer 1

The correct option is A

$x=\frac{-2}{\sqrt{3}},x=-\sqrt{3}$

Multiplying $\sqrt{3}$ by $2\sqrt{3}$ gives $6$

Factors of $6$ are

$1\times 6=6$ sum: $1+6=7$

$2\times 3=6$ sum: $2+3=5$

So, our required pair of factors is $2$ and $3$

$\Rightarrow \sqrt{3}{x}^2+5x+2\sqrt{3}=0$

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

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

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

$\therefore$ Factors are $(\sqrt{3}x+2),(x+\sqrt{3})$

$\Rightarrow (\sqrt{3}x+2)=0\text{and}(x+\sqrt{3})=0$

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