Question

Find the square root of $28-6\sqrt{3}$.
A. $\pm \left( \sqrt{3}-1 \right)$
B. $\left( 3\sqrt{3}+1 \right)$
C. $\pm \left( 3\sqrt{3}-1 \right)$
D. $\left( 3\sqrt{3}-1 \right)$

Answer 1

Hint: We have been given an equation of $28-6\sqrt{3}$. We need to find the square root form of the given. The possible forms are ${{\left( a+b \right)}^{2}}$ or ${{\left( a-b \right)}^{2}}$. Then from the part of $2ab$, we express in the form of ${{\left( a-b \right)}^{2}}$. Then we assume the a and b of the summation where $ab=3\sqrt{3}$ and ${{a}^{2}}+{{b}^{2}}=28$. We find values of a and b.

Complete step-by-step answer:
We need to find the square root of $28-6\sqrt{3}$.
Now we have to get the form of either ${{\left( a+b \right)}^{2}}$ or ${{\left( a-b \right)}^{2}}$.
The expansion of the squares is ${{\left( a\pm b \right)}^{2}}={{a}^{2}}+{{b}^{2}}\pm 2ab$.
In the given equation of $28-6\sqrt{3}$, we try to express 28 as ${{a}^{2}}+{{b}^{2}}$ and $6\sqrt{3}$ as $\pm 2ab$.
Now the sign before $6\sqrt{3}$ is negative. So, we will try to form the square of ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$.
As $2ab=6\sqrt{3}\Rightarrow ab=3\sqrt{3}$.
Now we get multiples of two digits as $ab=3\sqrt{3}$ and the sum of their squares is 28.
${{a}^{2}}+{{b}^{2}}=28$.
We can break $ab=3\sqrt{3}$ as 3 and $\sqrt{3}$, but sum of their squares is ${{3}^{2}}+{{\left( \sqrt{3} \right)}^{2}}=9+3=12$.
Now we break them as 1 and $3\sqrt{3}$, but sum of their squares is ${{1}^{2}}+{{\left( 3\sqrt{3} \right)}^{2}}=1+27=28$.
So, the terms are 1 and $3\sqrt{3}$. Without loss of generality we choose $a=3\sqrt{3},b=1$.
So, we form the square part as $28-6\sqrt{3}={{\left( 3\sqrt{3}-1 \right)}^{2}}$.
Now when we are taking the square root of the term, we have $\sqrt{28-6\sqrt{3}}=\pm \left( 3\sqrt{3}-1 \right)$.
So, the square root of $28-6\sqrt{3}$ is $\pm \left( 3\sqrt{3}-1 \right)$.

So, the correct answer is “Option D”.

Note: We can also use the formula of ${{\left( a-b \right)}^{2}}={{\left( a+b \right)}^{2}}-4ab$ to find the sum of difference of those two points. From there we can solve to find the values of a and b. Taking both positive and negative signs is mandatory as square root has two values.