Question

Find the value of $x - y$ when $x + y = 9$ and $xy = 14$

Answer 1

Hint- Form the identity of ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ by squaring both side of $x + y = 9$ and then putting value of $xy = 14$ wherever required.

Complete step-by-step answer:
Given,$x + y = 9$---(i)
 And $xy = 14$--- (ii)
To find-($x - y$ ) squaring both sides of eq. (i), we have
${\left( {x + y} \right)^2} = {\left( 9 \right)^2} \Rightarrow {x^2} + {y^2} + 2xy = 81$
On putting value from eq. (ii), we have
$\begin{gathered}
  {x^2} + {y^2} + 2\left( {14} \right) = 81 \Rightarrow {x^2} + {y^2} + 28 = 81 \\
  {x^2} + {y^2} = 81 - 28 = 53 \\
\end{gathered} $
Now we know that,${\left( {x - y} \right)^2} = {x^2} + {y^2} - 2xy$
$\begin{gathered}
   \Rightarrow {\left( {x - y} \right)^2} = 53 - 2\left( {14} \right) = 53 - 28 \\
    \\
\end{gathered} $
${\left( {x - y} \right)^2} = 25 \Rightarrow x - y = \sqrt {25} $ $ \Rightarrow x - y = \pm 5$
The Answer is $x - y = \pm 5$

Note: This type of question is generally solved by using identities like${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ . You just need to remember the identities and their result.