Question

If ${{a}^{2}}+{{b}^{2}}=41$ and ab = 4, then find the value of a – b:
A. $\sqrt{35}$
B. 6
C. $\sqrt{33}$
D. 7

Answer 1

Hint: Here we will use the algebraic formula of ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ . After that we are going to substitute all the given values and then we find the value of a – b by taking under root on both the sides using the above expression.

Complete step by step answer:
Let’s start our solution by first writing the formula that we are going to use,
${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
In the above formula we know all the required values that we need to find a – b,
Now we will substitute the values of ab = 4 and ${{a}^{2}}+{{b}^{2}}=41$ in ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$,
After substituting the values we get,
$\begin{align}
  & {{\left( a-b \right)}^{2}}=41-2\left( 4 \right) \\
 & {{\left( a-b \right)}^{2}}=33 \\
 & a-b=\sqrt{33} \\
\end{align}$
Hence, we have found the value a – b, which comes out to be $\sqrt{33}$ .
Hence, from this we can say that option (c) is correct.

Note: We have used ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$to find the value of a – b, as all the values were given in the question that we need to find the value of a – b, we can also use ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ to find the value of ${{\left( a+b \right)}^{2}}$ and then after finding that value we can use another formula which gives the relation between a + b and a – b, ${{\left( a-b \right)}^{2}}={{\left( a+b \right)}^{2}}-4ab$ we will use this formula to find the value of a – b, after we have found the value of a + b. This method is a bit longer than the one we have used.