If $3x + 5y = 11$ and $xy = 2$, find the value of $9x^2 + 25y^2$.

Given:

$3x + 5y = 11$ and $xy = 2$

To do:

We have to find the value of $9x^2+25y^2$.

Solution:

The given expressions are $3x + 5y = 11$ and $xy = 2$. Here, we have to find the value of $9x^2+25y^2$. So, by squaring the given expression and using the identity $(a+b)^2=a^2+2ab+b^2$, we can find the required value.

$xy = 2$............(i)

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

Now,

$3x + 5y = 11$

Squaring on both sides, we get,

$(3x + 5y)^2 = (11)^2$                 [Using (ii)]

$(3x)^2+2(3x)(5y)+(5y)^2=121$

$9x^2+30xy+25y^2=121$

$9x^2+30(2)+25y^2=121$                     [Using (i)]

$9x^2+60+25y^2=121$

$9x^2+25y^2=121-60$              (Transposing $60$ to RHS)

$9x^2+25y^2=61$

Hence, the value of $9x^2+25y^2$ is $61$.