Question

Which of the following operations could make the expression $x^2+\frac{8x}{\sqrt{3}}$ a perfect square?

• A. Add $\frac{64}{3}$
• B. Add $\frac{16}{3}$
• C. Add $\frac{64}{9}$
  
• D. Subtract $\frac{64}{3}$

Answer 1

The correct option is B

Add $\frac{16}{3}$

To make $x^2+\frac{8x}{\sqrt{3}}$ a perfect square, divide the co efficient of $x$ by 2 and the add the square of the result to make this a perfect square.

Hence $\frac{8}{\sqrt{3}}÷2=\frac{4}{\sqrt{3}}$

Square of ${\left(\frac{4}{\sqrt{3}}\right)}^2=\frac{16}{3}$

So, we should add $\frac{16}{3}$ to make $x^2+\frac{8x}{\sqrt{3}}$ a perfect square.

$\Rightarrow x^2+\frac{8x}{\sqrt{3}}+\frac{16}{3}={(x+\frac{4}{\sqrt{3}})}^2$