Question

$\text{Which of the following operations can}$
$\text{convert the expression}(x^2+\frac{8x}{\sqrt{3}})$
$\text{into a perfect square?}$

• A. $\text{Subtraction of}\frac{8x}{\sqrt{3}}$
• B. $\text{Addition of}{\left(\frac{8}{3}x\right)}^2.$
• C. $\text{Addition of}{\left(\frac{8}{\sqrt{3}}\right)}^2.$
• D. $\text{Addition of}{\left(\frac{4}{\sqrt{3}}\right)}^2.$

Answer 1

The correct option is D $\text{Addition of}{\left(\frac{4}{\sqrt{3}}\right)}^2.$

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

$\therefore \text{Adding}(\frac{8}{\sqrt{3}}÷2{)}^2=(\frac{4}{\sqrt{3}}{)}^2\text{to}{x}^2+\frac{8x}{\sqrt{3}},\text{we have,}$

$x^2+\frac{8x}{\sqrt{3}}+(\frac{4}{\sqrt{3}}{)}^2=x^2+2(x\left)\right(\frac{4}{\sqrt{3}})+(\frac{4}{\sqrt{3}}{)}^2=(x+\frac{4}{\sqrt{3}}{)}^2,\text{a perfect square}$

$\text{(Using the identity,}(a+b{)}^2=a^2+2ab+b^2)$