Question

Find the value of (a–b) if ab = 2 and ${a^2} + {b^2} = 85$.
$
  (a){\text{ }} \pm {\text{9}} \\
  (b){\text{ 9}} \\
  (c){\text{ - 9}} \\
  (d){\text{ }} \pm {\text{9}} \\
 $

Answer 1

Hint: In this question the value of ${a^2} + {b^2}$ is given along with the value of ab and we need to find the value of a-b. Use the algebraic identity that is ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$. This will help to get the answer.

Complete step-by-step answer:
Given data
${a^2} + {b^2} = 85$………………………. (1)
And $ab = 2$……………………………. (2)
Now we have to find out the value of (a - b).
Now it is a known fact that ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$……………………………. (3)
Now from equation (1) and (2) substitute the value in equation (3) we have,
$ \Rightarrow {\left( {a - b} \right)^2} = 85 - 2\left( 2 \right) = 85 - 4 = 81$
Now take square root on both sides we have,
$ \Rightarrow \sqrt {{{\left( {a - b} \right)}^2}} = \sqrt {81} $
$ \Rightarrow \left( {a - b} \right) = \pm 9$
So this is the required value of (a - b).
Hence option (a) is correct.

Note: Whenever we face such types of problems the key concept is to have a good gist of the basic algebraic identities, some of them are being used above. It is always advised to remember these identities as it helps to save a lot of time while solving such problems.