Question

Which of the following operations could help convert the expression $x^2+\frac{4x}{\sqrt{3}}$ a perfect square?

• A. By adding $\left(\frac{4}{3}\right)$
• B. By adding $\left(\frac{4}{3}x\right)$
• C. By adding $\left(\frac{8}{3}x\right)$
• D. By adding ${\left(\frac{8}{3}\right)}^2$

Answer 1

The correct option is A By adding $\left(\frac{4}{3}\right)$

$x^2+\frac{4x}{\sqrt{3}}$ is same as $x^2+2\times x\times \frac{2}{\sqrt{3}}$.

Adding $(\frac{2}{\sqrt{3}}{)}^2$ to $x^2+\frac{4x}{\sqrt{3}},$ we get,

$x^2+\frac{4x}{\sqrt{3}}+(\frac{2}{\sqrt{3}}{)}^2=x^2+2(x\left)\right(\frac{2}{\sqrt{3}})+(\frac{4}{3})$
$=(x+\frac{2}{\sqrt{3}}{)}^2,\text{a perfect square}$

$\left[\text{Using the identity}\right(a+b{)}^2=a^2+2ab+b^2\left)\right]$

Therefore, $\frac{4}{3}$ should be added to $x^2+\frac{4x}{\sqrt{3}}$ to make it a perfect square.