If $x – y = 7$ and $xy = 9$, find the value of $x^2+y^2$.

Given:

$x – y = 7$ and $xy = 9$

To do:

We have to find the value of $x^2+y^2$.

Solution:

The given expressions are $x – y = 7$ and $xy = 9$. Here, we have to find the value of $x^2 + y^2$. So, by squaring the given expression and using the identity $(a-b)^2=a^2-2ab+b^2$, we can find the required value.

$xy = 9$............(i)

$(a-b)^2=a^2-2ab+b^2$.............(ii)

Now,

$x – y = 7$

Squaring on both sides, we get,

$(x – y)^2 = 7^2$                 [Using (ii)]

$x^2-2xy+y^2=49$

$x^2-2(9)+y^2=49$                     [Using (i)]

$x^2-18+y^2=49$

$x^2+y^2=49+18$              (Transposing $-18$ to RHS)

$x^2+y^2=67$

Hence, the value of $x^2+y^2$ is $67$.