Question

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

• A. $Add(\frac{8}{\sqrt{3}}{)}^2$
• B. $Add(\frac{4}{\sqrt{3}}{)}^2$
• C. $Add(\frac{8}{3}x{)}^2$
  
• D. $Subtract\frac{8x}{\sqrt{3}}$

Answer 1

Answer: (B), (D)

The correct options are
B

$Add(\frac{4}{\sqrt{3}}{)}^2$

D

$Subtract\frac{8x}{\sqrt{3}}$

For making this expression a perfect square,
$x^2+\frac{8x}{\sqrt{3}}$
$=x^2+2\times \frac{4}{\sqrt{3}}\times x$ (here the $(\frac{4}{\sqrt{3}}{)}^2$ term is missing which is required to make the expression a perfect square)
So, We should add $(\frac{4}{\sqrt{3}}{)}^2$
$\Rightarrow x^2+2\times \frac{4}{\sqrt{3}}\times x+(\frac{4}{\sqrt{3}}{)}^2$
$\Rightarrow (x+\frac{4}{\sqrt{3}}{)}^2$
So if the term $(\frac{4}{\sqrt{3}}{)}^2$ is added, the expression $x^2+\frac{8x}{\sqrt{3}}$ becomes a perfect square.

Alternatively we can make the expression $x^2+\frac{8x}{\sqrt{3}}$ equal \(x^2\) (a perfect square) by subtracting the expression $\frac{8x}{\sqrt{3}}$.